Adjust oven rack to upper-middle position and heat broiler. Toss 30 ounces potatoes with 2 Tablespoons butter, 1 teaspoon salt, and ¼ teaspoon pepper in large bowl. Wrap tightly with plastic and microwave until potatoes are tender, about 7 minutes.
While potatoes are cooking, cook 1 pound beef and 1 chopped onion in large heatproof skillet over medium-high heat until beef is no longer pink, about 5 minutes. Drain beef mixture in colander.
Return drained beef mixture to pan. Stir in 1 ½ teaspoons thyme and ¼ cup flour and cook until incorporated, about 1 minute. Stir in 2 cups broth, 2 Tablespoons tomato paste, and 2 teaspoons Worcestershire. Cook until sauce is thickened, about 5 minutes. Stir in 2 cups pea and carrot medley and simmer until heated through, about 1 minute. Season with salt and pepper.
Scatter cooked potatoes over beef mixture and brush with remaining butter. Broil until potatoes are golden brown, 3 to 5 minutes. Serve.