In the bowl of a stand mixer fitted with a dough hook attachment, place the yeast, brown sugar, salt, water, oil and 3 cups of flour. Set the speed to low and let it go until the dough starts to come together. Continue to let it knead until the dough forms a soft ball that pulls away from the sides of the bowl. It should be tacky, but shouldn't really stick to the skin. Add in more flour if you need to, to get it to that point in 1/4 cup increments. I usually end up using close to 3 3/4 cups.
Place the dough into a lightly greased bowl and place a clean towel over it. Let it rise for 1 hour, until double in size. About halfway through rise time, go ahead and start the marinara sauce and the sausage filling. For the marinara, get the oil heating over medium heat in a small pot. Once it's hot, add the garlic and cook until just fragrant. Then stir in the tomato sauce, brown sugar and seasonings. Once it begins to bubble, lower the heat and let it gently simmer until calzones are done.
For the sausage filling, heat the oil over medium heat. Once it's hot, add the sausage and cook, breaking it up as you go, until it is browned throughout. Then add the peppers and onion and cook until they have softened up a bit. Add the garlic and seasonings and cook just another couple of minutes before removing the pan from the heat and setting it aside. Preheat the oven to 400F and line a large pan with parchment paper.
Once the dough is done rising, turn it out onto a large table or counter. Seperate it into 4 equal sized balls. One at a time, roll each ball out into a 12" circle. Once it's rolled out, place 1/4 of the sausage mix into one side of the circle, leaving room along the edges. Then sprinkle 1/4 cup each of white cheddar and mozzarella over the meat. Stretch the other side of the dough circle over the meat and cheese and pinch the two sides closed. Then fold and tuck the ends firmly together.
Repeat with each of the dough balls. Place all calzones on the prepared baking pan and use a sharp knife to cut 3 slits in each calzone. Then brush the beaten egg all over each one. Bake for 15-20 minutes, until they are golden brown. Then remove the marinara from the heat. Let the calzone sit for just a few minutes before serving.
