Preheat the oven to 425 degrees F. Remove the stems from 4 of the mushrooms and set the stems aside. Put the 4 stemmed mushroom caps, gill-side-up, in a large baking dish, making sure there is at least 1/2 inch of space between them. Drizzle with 2 tablespoons of the oil. Season with salt and pepper. Roast until a knife inserted in the center meets just a bit of resistance, 20 to 25 minutes. Remove and keep warm.
Clean and roughly chop the reserved mushroom stems and the remaining portobello. Heat the remaining 2 teaspoons oil in a medium nonstick skillet over medium heat. Add the onion and a pinch of salt and cook, stirring, until soft and translucent, about 4 minutes. Add the chopped mushrooms, garlic and a generous amount of black pepper; cook until the stems are soft, about 4 minutes.
Sprinkle the flour over the onions and mushrooms and stir until all of the flour has been absorbed. Stir in the beef broth. Bring to a simmer and cook, stirring, until thick, about 5 minutes. Remove from the heat and stir in the sour cream, parsley, mustard and Worcestershire sauce. . Keep warm while you grill the burgers.
Shape the ground beef into four patties, each about 5-inches in diameter. Sprinkle well with salt and pepper. Heat a nonstick grill pan over medium heat. When the pan is hot, spray it liberally with nonstick cooking spray. Add the burgers and cook until nicely browned, about 2 minutes. Flip the burgers and cook for another 1 1/2 minutes for medium-rare.
Put the bottoms of the buns on serving plates. Top each with a lettuce leaf and a roasted mushroom, gill-side down, then top with a burger and a generous dollop of the sour cream-mushroom sauce. Cover with bun tops and serve.
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