In the bowl of a stand mixer, combine yeast, brown sugar, salt, water, olive oil, and flour. Knead on low until dough starts to come together. Continue kneading until the dough forms a nice, soft dough ball and the sides of the bowl are clean.
Let rise in a lightly greased bowl, covered, for 1 hour, until doubled in size.
Meanwhile, make the filling. Heat a large heavy skillet over medium-high heat. Add in sausage and cook 5 minutes. Use a spatula to break up the sausage into crumbles. After 5 minutes, add in the bell peppers and mushrooms. Continue to saute until vegetables are tender crisp and sausage is cooked through, about 5 minutes.
Divide the dough into 4 equal pieces. Roll into circles with a 12-inch diameter. Spoon the sausage mixture onto one half of the dough, leaving a 1 inch edge. Top with 1/2 cup cheese per calzone. Fold the dough over the filling and seal the edges.
Transfer stuffed calzones to a lightly greased baking sheet. Use a sharp knife to make three slits in the top of each calzone to allow steam to vent.
Bake in a 400 degree oven for 15 to 20 minutes until bread is golden brown. Serve hot.