For me, planning an Asian vegetable garden is a bit more complicated than planning a regular vegetable garden.
With a regular vegetable garden, I pretty much know what vegetables I like so I know the majority being grown I'll plant more than I need, cause I'll use them for preserving and freeze some.

Although I've heard of most Asian vegetables, I've never used them in cooking, nor have I tasted them. I don't have any exposure to most of them. As a result, the first Asian vegetable garden I grow, I'll just be growing enough
to eat fresh, to see what I like, what I don't, etc.
I also don't know which Asian vegetables freeze well, store well, etc.

I'd need to know how many I could grow for one person that would be enough for me just to eat fresh. Please specify the quantities with how many of each plant I'd plant, and NOT by weight.

The veggies I'd try to grow for an Asian veggie garden are listed below:

Daikon

Kabocha (Japanese pumpkin; much smaller than American pumpkins)

Winter Melon
Pickling Melon
Sword Beans
Long beans
Amaranth (water spinach)
Chinese celery
mustard greens
mizuna
mibuna
burdock root (gobo; it's a long root veggie)
bitter gourd
Asian Eggplants (Chinese, Thai, and Japanese varieties)
Bok Choy and Pak Choy (pak choy is baby bok choy)
Misome
Turnip Greens
Watercress
Komatsuna
Azuki Beans, Black soybeans (kuro mame)

Sesame seeds, black, white (don't know about this; all depends on if my growing season is long enough)

Herbs:

Yomogi (mugwort, Japanese herb)
Shisho (Japanese variety of Basil)
Thai Basil
Garlic Chives
Bird's Eye Chilies

Daikon: I grew a short row but wasted most of the crop because it bolted sooner than I expected. I did not have a good set of recipes so I didn't use as much as I might this coming season. I also plan to plant Chinese Watermelon or Red Meat Radishes. I like Daikon and carrot pickles but my family does not.

Kabocha (Japanese pumpkin; much smaller than American pumpkins) We grew a variety from Johnny's called Confection which I believe is from this family. It's a winter squash. Confection has been a great keeper, stored in our semi-heated garage. It is March and we still have some good squashes. Our crop wasn't as good the second year as the first but we had a very rainy summer. I discovered we much prefer using the squash in Massaman curry instead of sweet potatoes (which we do not grow).
I'm not familiar with the next 3:
Winter Melon
Pickling Melon
Sword Beans
Long beans: I am trying these this year. I planted too late last summer and the beans didn't have enough time to mature before our early frost. Check the days to harvest.
Amaranth (water spinach): not familiar
Chinese celery: I take this to mean cilantro. I have a hard time growing it because it bolts early. The seeds sown late did better in cooler weather.
mustard greens: we discovered we don't like mustard greens
mizuna
mibuna
burdock root (gobo; it's a long root veggie)
bitter gourd
Asian Eggplants (Chinese, Thai, and Japanese varieties)
Bok Choy and Pak Choy (pak choy is baby bok choy): These bolt easily. Just grow a few plants so you can use them fresh, and do successive plantings.
Misome: not familiar
Turnip Greens: This year I will grow choy sum for the first time. I finally bought some at the Asian grocery. We like it better than Swiss chard.
Watercress: not familiar
Komatsuna:not familiar
Azuki Beans, Black soybeans (kuro mame): I don't grow

Sesame seeds, black, white (don't know about this; all depends on if my growing season is long enough): cheaper to buy

Herbs:
Yomogi (mugwort, Japanese herb)
Shisho (Japanese variety of Basil) This really isn't a basil but it's related. I don't use it much.
Thai Basil: I love the flavor of this. A small patch would be good, maybe 3 x 1'. I did not freeze or dry this.
Garlic Chives: I really haven't made good use of these although I have grown for many years. A short row is probably sufficient.
Bird's Eye Chilies: I grow one plant but we don't use many hot peppers. I use bottled sauces that add enough heat.

Amaranth and Shiso are both sprouting in my garden now. Amaranth is not really water spinach. Water Spinach does not have much flavor of its own. While Amaranth does. It's high in Iron so you'll like it if you like true Spinach. Amaranth may grow tall, while Water Spinach will spread horizontally.

By Shiso, I guess you meant it looks like basils because Shiso is actaully related to the Mint family. Sometimes I plant Mint in pots to entend their season. I imaging moving Mint pots may be worth the effort in your area because Shosi may be hard to find fresh in your local stores.

Mustard green and Bok Choy should be great choice for your zone because they do well in our winter. I'm not sure about soy bean or eggplant because mine did poorly for me last year due to the early cool summer in my area, and we're in zone 8.

I forgot to mention Garlic Chive is a winner for me because I only need to plant them once, and divide every five year. They survive heat and hail. If I have to move when I live to be 100, I will dig up my garlic chives and transplant them where ever i go.

Daikon. These are a pain in the butt to dig out so make sure you have a good bed of garden soil. I grow these in raised beds for that reason.

Like another had suggested, how about adding lemon grass? Easy to grow stuff. Next time you are in an Asian market buy a stalk which still contains it's root ball. Place it in water for a few days in a sunny window and it will start to root. Plant that and in a few weeks you will have a nice size mound of Lemon Grass.