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 oil over medium-high heat. Add the mushrooms and cook, stirring occasionally, until they are beginning to brown, about 5 minutes. Add the onions and red peppers, season with the oregano and 1/8 teaspoon salt and continue cooking, stirring occasionally, until the onions and peppers are softened and beginning to brown, about 8 minutes. Turn off the heat and add the tomatoes and 1/8 teaspoon 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. Butter the sides of each bun and grill or broil until golden, about 1 minute. Place each hot dog in a toasted bun. Divide the vegetable-cheese mixture evenly over the dogs.