Heat oven to 450 degrees F.
Peel the onion half and thinly slice it. Place the slices in a 2-quart, microwave-safe dish. Add the mushrooms and pepper slices (optional) to the dish with the onions. Cover the dish with microwave-safe plastic wrap and vent one corner. Microwave, on HIGH until the onions and peppers are wilted and the mushrooms have released their liquid, about 3 minutes.
While the vegetables cook, place the pizza crust on an ungreased baking sheet.
Stir together the mustard and sour cream and spread the mixture evenly over the crust.
The chicken and ham pieces should be no larger than bite-size, so cut them if necessary. Sprinkle the chicken and ham pieces evenly over the crust.
Remove the vegetables from the microwave and drain away any liquid from the dish. Distribute the vegetables evenly over the pizza. Sprinkle the cheese evenly on top. Bake, uncovered, until the pizza begins to brown at the edges, 10-12 minutes. Cut into 4 wedges and serve.