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Brown the beef: In a large skillet over medium heat, cook ground beef with ½ tsp salt, ¼ tsp black pepper, and ½ tsp garlic powder until browned. Remove from skillet and set aside.
Cook the veggies: In the same skillet, melt butter. Add onion, grated carrot, and garlic. Sauté for 3 minutes until softened.
Add broccoli: Stir in chopped broccoli florets and cook for 4–5 minutes, until slightly tender.
Make the sauce: Return beef to skillet. Pour in milk and heavy whipping cream. Simmer for 5–6 minutes, stirring occasionally, until slightly thickened.
Season it up: Stir in Dijon mustard, ½ tsp salt, 1 tsp garlic powder, 1 tsp smoked paprika, and ¼ tsp black pepper.
Melt the cheese (low & slow): Reduce heat to low. Add cheddar and Colby Jack in small handfuls, stirring after each addition until completely melted and smooth.
Serve & enjoy: Spoon the cheesy broccoli beef mixture over bowls of hot cooked rice or pasta, garnish with extra cheddar cheese, and serve immediately.