Heat the oven to 375°F. Grease 8 cups of a standard muffin tin.
Put the potatoes in a saucepan and cover with cold water. Bring to a boil, then reduce the heat and simmer until the potatoes are fork-tender, about 15 minutes.
While the potatoes cook, combine the ground beef, breadcrumbs, egg, 1/4 cup milk, 1 tablespoon ketchup, Worcestershire sauce, garlic powder, salt, and black pepper in a bowl. Mix gently just until combined.
Divide the meat mixture into 8 portions. Press half of each portion into the muffin cups, leaving a small hollow in the center. Place a cheddar cube in each hollow, then cover with the remaining meat mixture and seal the edges around the cheese.
Brush the tops with the remaining 1 tablespoon ketchup. Bake until the meat is cooked through, about 20 to 25 minutes.
Drain the cooked potatoes well and return them to the saucepan. Add the butter, sour cream, and 2 tablespoons milk. Mash until smooth, then stir in the chopped chives.
Let the meatloaf muffins rest for 5 minutes, then lift them from the tin. Spoon or pipe the chive mashed potatoes onto each muffin and serve warm.