Preheat the oven to 425º F
Halve the peppers through the stems; remove the stems, seeds and membranes. In a 9x13 baking dish, toss the peppers with 1 tablespoon of the olive oil and 1/2 teaspoon salt, then arrange the peppers cut-sides up. Set aside.
Heat the remaining 3 tablespoons olive oil in a large skillet over medium heat. Add the hamburger and cook, breaking it up, until no longer pink, about 3 minutes. Add the carrots, onions, mushroom and garlic and cook until the vegetables begin to soften, about 5 minutes Season with the cumin, smoked paprika, 1/2 teaspoon salt and some pepper; stir to combine. Add the tomatoes, ketchup, mustard, brown sugar, vinegar, parsley and 3/4 cup water. Bring to a rapid simmer and cook until thickened, about 10 minutes.
Spoon the filling into the pepper halves and sprinkle with the Cheddar. Pour 1/4 cup water into the bottom of the baking dish and tent with foil (make sure it doesn't touch the cheese.). Bake until the peppers begin to soften, 20-25 minutes. Uncover and bake until peppers are very tender and the filling is browned and bubbly, another 15-20 minutes.
Serve the stuffed peppers topped with crumbled corn chips.