Preheat the oven to 375 degrees F. Heat 1 tablespoon butter in a large skillet over medium heat. Add the ground beef and cook until browned, breaking it up with a spoon.
Stir in the onion, carrots, and garlic. Cook for 5 minutes until the vegetables soften. Add the frozen peas, season with salt and pepper, and cook for 2 more minutes. Remove from heat.
While the filling cooks, bring a medium pot of water to a boil. Add the cauliflower florets and cook for 8 to 10 minutes until tender. Drain well.
Return the cauliflower to the pot. Add the remaining 2 tablespoons butter, milk, and a pinch of salt. Mash or blend until smooth.
Spoon the beef mixture into a baking dish and spread it evenly. Top with the mashed cauliflower and sprinkle the cheddar cheese over the surface.
Bake for 15 minutes until the cheese melts and the edges bubble. Let it rest for 5 minutes before serving.