Rinse mushrooms, and trim and discard discolored stem ends; thinly slice mushrooms. Peel onion and thinly slice. Rinse, stem, seed, and thinly slice bell peppers. Peel and mince garlic.
Crumble sausages into a 5- to 6-quart pan over high heat; stir until no longer pink, 4 to 6 minutes. Add mushrooms, onion, bell peppers, and garlic; stir until vegetables begin to brown, about 15 minutes. Remove from heat and stir in marinara sauce; add salt and pepper to taste. Scrape into a shallow, round 1 1/2-quart casserole or 9- to 10-inch pie pan that holds at least 6 cups. Sprinkle evenly with mozzarella.
Separate breadstick dough into individual strips. Gently twist each strip and stretch it to fit from rim to rim over filling (if necessary, pinch ends of strips together to make longer or cut to fit), laying half the strips about 1 inch apart over pie in 1 direction and half at right angles on top of the first. If desired, weave strips over and under each other in a lattice pattern. Fold excess under and press ends against rim.
Brush dough strips lightly with oil. Sprinkle pie evenly with parmesan cheese. Set pie on a foil-lined 12- by 15-inch baking sheet.
Bake in a 400° regular oven or a 375° convection oven until crust is browned and filling is bubbling, 20 to 30 minutes. If edges brown too rapidly, fold foil from baking sheet up to cover pie rim loosely.