Place peeled potatoes in a large pot, cover with salted water, and bring to a boil. Cook for 15-20 minutes until fork-tender.
While potatoes cook, melt 4 tablespoons of butter in a small saucepan over medium heat. Swirl constantly until the butter foams and turns a golden brown color with a nutty aroma (about 3-5 minutes). Remove from heat immediately.
Drain potatoes and return to the pot. Mash with the brown butter, warm milk, and 3/4 cup of the grated Manchego cheese. Season with salt and pepper. Set aside.
Preheat oven to 400°F (200°C).
In a large oven-safe skillet or Dutch oven, heat the remaining 2 tablespoons of butter over medium-high heat. Add sliced mushrooms and cook undisturbed for 4-5 minutes until browned. Remove mushrooms and set aside.
In the same skillet, add the ground beef. Break it up with a spoon and cook until browned. Drain excess fat if necessary.
Add the diced onion, carrots, and red bell pepper to the beef. Sauté for 5-6 minutes until vegetables soften. Add minced garlic, smoked paprika, cumin, and thyme; cook for 1 minute until fragrant.
Stir in the tomato paste and cook for 1 minute. Pour in the beef broth and return the cooked mushrooms to the pan. Simmer for 3-5 minutes until the sauce slightly thickens.
Spread the mashed potatoes evenly over the beef filling. Sprinkle the remaining 1/4 cup of Manchego cheese on top.
Bake for 20 minutes until the filling is bubbling. Broil for the last 2-3 minutes to golden brown the top. Let rest for 10 minutes before serving.