Heat a 10-inch saute pan over medium heat. Cook the hot dogs, browning them on all sides, until warmed through, about 5 minutes. Remove to a plate and keep warm while making the topping.
In the same pan, heat the olive oil over medium-high heat. Add the mushrooms to the hot pan and cook, stirring occasionally, until they begin to brown, 5 minutes. Add the onions and red peppers, and season with half of the salt and the oregano. Continue cooking, stirring occasionally, until the onions and peppers soften and begin to brown, about 8 minutes. Turn off the heat and add the tomatoes and the remaining salt. Stir with a rubber spatula just to combine and sprinkle the mozzarella on top to melt.
Preheat a grill pan or a broiler to medium-high heat. Spread some butter on each bun and grill or broil until golden, 1 minute. Place each dog in a toasted bun. Divide the pepper and cheese mixture evenly over the dogs, and serve.