The same way you would make a large breed out of a small one, by out crossing to the different sized breed.
I started making large fowl Crele Old English by crossing an excellent Crele bantam cock onto a a pair of Large BB Red old English hens. From there I have kept breeding for size and color. You have to raise a lot and keep them till mature, especially the cockerels. I am in year 5 and raised 75 this year. I culled down to 4 stags and 12 pullets so far.
To make bantam American Games I would start with the large male to the bantam female. Use as close of a color pattern as possible. Buy a scale and way them, looks can be deceiving
Bob