it sounds like you're looking for a double-flowered semperflorens. B. semperflorens are the types used as landscaping annuals, and they also come in several double-flowered varieties. I grow some as houseplants.

The "Proven Winners" growers, who distribute to many garden centers, have a "new" series they call "Gumdrop". You can search on "Gumdrop series Begonias" to see images.

I know a couple of mail-order places that carry double flowered semps. One is Taylor Greenhouses in Western New York State. She has many types of begonias, including the double flowered semps.
www.taylorgreenhouses.com/

Also, Pepper's Greenhouses in Delaware has several of them.
www.accentsforhomeandgarden.com/

They occasionally show up at garden centers, and I've even stumbled across them at a big box store (not labeled as to what they were, though).

Good luck in your search!
-Jane