Cook ground beef in a large nonstick skillet over medium-high heat, stirring until it crumbles and is no longer pink. Remove from skillet, and drain well. Wipe skillet clean with a paper towel.
Melt butter in skillet over medium-high heat. Add bell pepper strips and sliced mushrooms, and sauté 8 minutes or until mushrooms are lightly browned and liquid evaporates. Remove from heat; stir in beef, 1 jar pizza sauce, and cheese.
Unroll each can of pizza crust dough on a lightly floured surface; cut each crust into quarters. Press each dough quarter into a 7- x 6-inch rectangle. Spoon about 2/3 cup meat mixture in the center of each rectangle. Fold 1 short side of dough over filling, pressing or crimping edges to seal. Place beef pockets on 2 lightly greased baking sheets, and cut slits in tops to allow steam to escape. Brush lightly with olive oil.
Bake, on separate oven racks, at 375° for 25 to 30 minutes or until golden. (Switch places of baking sheets in oven after 15 minutes if necessary for even baking.) Serve with pizza sauce, if desired.