Place a large skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Split each of the hot dogs open lengthwise without cutting all the way through, and lay them cut side-down in the hot skillet (like a book). Cook until golden brown and crispy, 5-6 minutes.
While the hot dogs are cooking up, place a medium pot over medium-high heat with the butter. When the butter has melted, sprinkle in the flour and cook for about 1 minute. Whisk in the milk, ketchup and mustard. Add the chili powder, some salt and pepper, and bring up to a bubble. Simmer until thickened, 2-3 minutes. Remove the pot from the heat and stir in the cheese until melted. Reserve warm.