Preheat oven to 350°F. In a large bowl stir together eggs, bread crumbs, mushrooms, carrot, onion, the 3 tablespoons ketchup, the mustard, Worcestershire sauce, and garlic. Add ground beef; mix well.
Lightly press about 1/3 cup of the meat mixture into each of twelve 2-1/2-inch muffin cups. Place muffin pan on a large baking sheet. In a small bowl stir together the 1/2 cup ketchup, the cranberry sauce, and chili powder. Top each meat portion with 1 tablespoon of the ketchup mixture.
Bake, uncovered, about 40 minutes or until internal temperature registers 160°F on an instant-read thermometer.
Meanwhile, heat potatoes according to package directions. Using a 1/4-cup ice cream scoop, top each muffin with potatoes. Sprinkle potatoes evenly with cheese. Return to oven and bake, uncovered, about 5 minutes more or until cheese melts.
Let the muffins stand in pan on a wire rack for 10 minutes. Using a slotted spoon, transfer muffins to a serving platter. If desired, sprinkle tops with snipped fresh chives.