In large, deep 12-inch skillet over medium heat melt butter. Add carrots and onion and cook until softened and edges of onion begin to brown, about 6 minutes. Stir in tomato paste, garlic, sugar, and 1/2 tablespoon Worcestershire sauce and cook until fragrant, about 1 minute. Increase heat to medium-high and add ground beef. Use wooden spoon to break up meat and cook until browned, about 5-6 minutes.
Stir in milk and cooking wine, scrapping up brown bits from bottom of pan. Cook mixture until liquid is nearly evaporated, about 14 minutes. Stir in crushed tomatoes and bring to a simmer. Continue to simmer until slightly thickened, about 2 minutes. Add remaining ½ tablespoon Worcestershire sauce, season with salt and pepper, and serve over sliced rolls. Garnish with freshly grated Parmesan cheese and serve.