In a skillet over medium heat, heat 2 tablespoons olive or vegetable oil. Add vegetables and garlic, and season with salt and pepper, cook to tender then cool completely.
Meanwhile, boil potatoes in salted water until tender.
Combine ground beef with 3 tablespoons Worcestershire, salt, pepper and cooled vegetables. Form 4 patties and drizzle them with oil.
Heat griddle or cast-iron skillet over medium-high heat. Cook burgers 10 minutes, turning once for medium doneness.
Switch broiler on high.
Drain potatoes and mash with milk, sour cream, horseradish, egg yolk, 2 tablespoons butter, chives, salt and pepper.
Top the burgers with mounds of potatoes and set under broiler to brown, in middle of oven, 3-4 minutes.