Pierce the potatoes with a fork in a few spots. Put on a microwave-safe plate and microwave until tender, 14 to 16 minutes.
Meanwhile, melt 1 tablespoon butter in a large skillet over medium-high heat. Add the onion, garlic, thyme and 1/4 teaspoon salt. Cook, stirring, until the onion is soft, about 4 minutes. Add the beef, 1/2 teaspoon salt and a few grinds of pepper and cook, breaking up the meat, until no longer pink, about 4 minutes.
Stir in the beef broth, ketchup and Worcestershire sauce and bring to a gentle simmer. Cook, stirring occasionally, until slightly thickened, 6 to 8 minutes. Stir in the peas and carrots and warm through, about 1 minute; season with salt and pepper.
Preheat the broiler. Halve the potatoes lengthwise and fluff the flesh with a fork; top each potato half with 1/2 tablespoon butter and season with salt and pepper. Place on a baking sheet. Divide the meat mixture among the potato halves, then top with the cheese. Broil until melted, about 2 minutes.
Photograph by Ryan Dausch