Put the potatoes in a medium saucepan and cover with cold water; season with salt. Bring to a boil and cook until tender, 25 minutes.
Meanwhile, combine the ground meat, panko, egg, onion, thyme, Worcestershire sauce, 1 teaspoon salt and a few grinds of pepper in a bowl; mix with your hands. Form into four 1/2-inch-thick oval patties.
Heat 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the patties; cook until browned, 3 minutes per side. Scatter the carrots and celery around the patties; cook until they start softening, 3 minutes. Reduce the heat to medium. Add the broth, cover and simmer until the patties are cooked through, 5 minutes. Remove the patties and increase the heat to high. Cook until the broth is slightly reduced, 2 minutes, then add 1 tablespoon butter and swirl until melted.
Drain the potatoes; mash with the half-and-half, the remaining 2 tablespoons butter, and salt and pepper to taste. Serve with the patties and vegetables.
Meatloaf mix is a combination of ground beef, pork and veal; you can also use a mix of 12 ounces ground chuck and 8 ounces ground pork.