I grew a garden in heavy clay soil when I lived in eastern WI a few years back. I found that raised beds were the way to go. Amend your soil with lots of organic matter, such as aged manure from a farm or horse stable or see if there is some kind of leaf/yard waste composting facility run by a city or county nearby. Clay can be hard to work at first but once you get it amended & built up, it will produce a great veg. garden. Also might be a good idea to get a soil test for pH & proper fertilizing.

Composted manure does wonders for all types of soil.
Try growing zucchini, broccoli, egglpant, and even some winter squash if the short summers of zone four allow it.

We have clay soil where I am at. I have a mushroom farm near the house that gives away compost, I also compost my yard and kitchen scraps. I add a few bags of potting soil every year also. After 3 years the raised beds soil is getting very nice. Now that I have a good base of organic material in the soil I plan on adding sand. Sand and clay turn to cement so wait till you have some nice broken down soil to add it to. I grow eggplant in pots and save the space in my raised beds for Cukes, tomato, salad greens, Italian peppers, squash and onions. Take your time working with the soil. Grow peas and beans in the fall and add them to the soil this will help alot. I can grow peas all winter in California. they grow like weeds, Try sugar snaps.

I grow lots of things on extremely heavy clay. It's hard work getting it in shape, but at least here, it has lots of nutrition for plants once you get it draining.

Lasagna gardening is excellent for clay, but it's slow. Basically, you lay layers of organic material over your area, up to two or three feet deep, mulch it, and let it "cook" for several months. Then you plant right into it. If you are in a hurry for this year, you'll need to try something else. (Keep it in mind for starting another bed or two come fall, though.)

Organic material is the answer. You will need a good four to six inches of whatever you can get your hands on (any combination): leaves, grass clippings, compost, manure, sawdust (add extra nitrogen to it the first year if its fresh), etc. Till it in to a depth of 8 to 12 inches. Then build your bed around it. Top it off with good top soil, composted manure, etc. and plant. As several said above, add more organic material to it each year and work it in. The work gets MUCH easier each year after the first. If you don't do the hard work now you'll be disappointed in your yields, and you'll have to do the hard work later. It's easier to do it BEFORE you get your raised beds built.

(I'm not potager expert, I just lurk here and admire the art! But I did start with some ugly clay dirt, and have some nice loamy, ferile soil now!).

We have very clay soil as well, and pretty nutrient-deficient. I went the lasagna route (tons of info in the soil forum), laid cardboard right on top of existing grass, and I piled finished compost on top-- a few inches. Planted right into that, right away (not perennials, not shrubs or bushes), and let it go. I had GREAT results the first year, and now all traces of cardboard & grass are gone, and it's 'shrunk' back to a normal-looking level, and the soil is just beautiful and great, as far down as I've seen. I now expand each year using this method, and am really happy with it. :) I'll never till again.

We've got very heavy clay soil here in central Texas as well. The soil looks suitable for making pots and bowls...

I am in my second year of amending the soil, mainly through the use of generous amounts of compost mixed in with a tiller. Last year my garden yields were not so great, in part because I got a late start (due to the date we moved into the house) and in part because of record drought. My best harvests last year were okra and basil; most other plants limped along feebly through the > 100 F heat. This year has been much, much better. I had a cover crop, and that also helped break up the heavy soil. One part of the garden got planted with a cover crop mix of vetches, oats, and bell beans in September, and this got tilled under in January. The other part got planted in January with fava beans, and the fave beans were harvested and pulled out last month.

This year, in addition to tilling compost in, I have also made small holes or trenches of compost in which I have directly planted the seeds or transplants, to give them extra nutrients as they start.

This year the amended garden has been yielding very well. The soil texture is better than last year, though it will probably take another couple of seasons to achieve the nice "chocolate cake" type soil that is possible after amendment. My clay + compost garden has far out-performed my trial 4' x 4' square garden raised bed, probably because the clay provides nutrients not present in the mix used in the raised bed. I am continuing to experiment with the raised beds as well by adding more compost and organic fertilizer.