To make the cheddar cheese beer sauce, add the beer to a sauce pan. Bring it to a boil. Toss the flour with the cheese. Once the beer is boiling, add the cheese and flour mixture, dial down the heat, and stir constantly. Toss in some salt and pepper. Stir, and add to a serving bowl.
Make the dough by adding the flour, yeast, salt, and sugar. Mix. Add the water, and with your wooden spoons, or hands, mix until blended. I use my KitchenAid for this process to keep everything in the bowl during the rising process. Mix for about 30 seconds. The dough should be a bit stiff. Cover the bowl with a towel, and let it sit at room temperature until the dough has doubled in volume, about 2 hours. You can do this ahead of time as well, and place in an oiled container, big enough to house the dough, and place in the refrigerator overnight.
Add the brats to medium sized pot. Toss in half of the sliced onions and then entire beer, and bring this to a simmer, cooking for about 25 minutes. Remove two brats, and slice into about 1/2 inch rounds. Set aside.
Preheat your oven to 500 degrees, and place your pizza stone, or sheet pan in the oven to preheat. Once preheated, let the stone stay in the oven for about 10 minutes to get nice and hot.
Once the dough has risen, divide into two portions. With one of the portions, add some flour to a large pastry board, and take that portion of the dough, and form into a large enough piece to place onto your pizza stone.
Place the dough on the preheated stone, then add the pizza sauce, making sure you get a nice, thin coating around the entire pizza dough.
Top with the shredded mozzarella cheese, then scatter half of the thinly sliced onions on top of the cheese. Place the sliced brats onto the pizza, then drizzle the cheddar cheese beer sauce on top of the pizza.
Place into the preheated oven and cook for about 25 minutes, or until the cheese is nice and bubbly and the crust is a dark golden color.
Remove from the oven and place onto a pizza sheet, shower the pizza with the chives, and let the pizza rest for about 10 minutes before slicing into it.
OK, the result is truly an awesome pizza. At first glance one might think it is just a sausage pizza, but then you get that awesome aroma from the beer brat. Biting into it yields everything you would want in a great brat, and then the onion, and that cheddar cheese beer sauce that melts right into the pizza, need I say more!? This pizza is a real winner, and I am glad that I can introduce you to some of the fine things that Wisconsin has to offer. I hope you enjoy!