To prepare the peppers, slice the tops off and remove any veins or seeds inside. Do not discard tops as you will use any remaining pepper attached to the stem.
Add peppers to a casserole dish, fill with about ½ inch of water, cover with aluminum foil, and bake in a 425o F oven for 20 minutes in order to steam, or parbake, the peppers.
While the peppers are steaming, cook the ground beef, onion, mushrooms, and any remaining pepper from the stem piece (diced) in a large saute pan over medium high heat until the meat is crumbly, about 10-15 minutes. Remove from heat and set aside.
In a separate pot, melt the butter and flour together over medium heat while stirring frequently to make a roux. After a few minutes when it has turned golden brown and has a nice aroma, whisk in the milk and continue to heat until mixture is thoroughly mixed and heated through. Remove from heat and stir in provolone and parmesan until melted.
Once peppers are done, remove from oven, discard water, and return peppers to casserole dish. Fill each pepper cavity with the ground beef filling until all of the peppers have been evenly filled. Top with provolone cheese sauce. If making ahead, store in airtight container in the refrigerator until ready to heat and serve.
Return peppers to oven and broil on low until the cheese turns a bubbly golden brown. This should only take a few minutes.