1. Preheat oven to 450 degrees. In a pot of salted water bring potatoes, carrots, and 3 cloves garlic to a boil. Reduce to a rapid simmer; cook until vegetables are tender, about 15 minutes. Drain and return to pot. Using a masher, mash vegetables until mostly smooth; stir in 3 tablespoons butter. Season with salt and pepper. Finely chop remaining 2 cloves garlic; set aside.

2. In a large skillet over high, melt remaining tablespoon butter. Add beef and brown, about 5 minutes. Remove meat from pan, leaving drippings. Reduce heat to medium-high, add onion, and cook until translucent, about 5 minutes. Add mushrooms and cook, stirring occasionally, until golden, about 10 minutes. Add chopped garlic, and cook until fragrant, about 1 minute. Return meat and collected juices to pan and stir in tomato paste. Cook, stirring, about 2 minutes. Stir in Worcestershire sauce and 1 cup water and bring to a boil. Reduce heat and simmer until liquid is slightly thickened, about 5 minutes. Season with salt and pepper.

3. Transfer meat mixture to a 9-inch square baking dish. Dollop mashed vegetables over meat and spread to edges of baking dish. Bake until juices are bubbling and topping is golden brown in spots, about 20 minutes. Let sit 5 minutes before serving.