For the onions, heat a few inches of oil in small pot over medium-high heat. While that heats up, cut 2-inch thick slices of onion and separate into 4 large rings, reserve remainder of onion for another use. Place buttermilk in a bowl and soak rings a few minutes in the buttermilk. Dredge rings in flour mixed with chives and lots of salt and coarse pepper, then dip them again in the buttermilk and again in the flour. Fry 1 or 2 rings at a time about 4 minutes until deep golden and cool/drain or rack.
Combine meat with Worcestershire, lots of pepper and a little salt. Form 4 large patties and coat with EVOO. Heat a large skillet over medium-high heat, add burgers and cook 4 minutes on each side, remove to platter and rest. Add a little more EVOO to pan and the butter, reduce heat a bit and saut mushrooms a few minutes. Add shallots, garlic and thyme, season with salt and pepper, and cook a couple of minutes more. Sprinkle a little flour around pan, stir then add sherry to deglaze. Add consomm and thicken. Stir in sour cream to combine.
Place lightly buttered toast on dinner plates and top with burgers. Pour warm stroganoff sauce over top and garnish with 1 large onion ring and parsley. Serve with watercress salad with Dijon vinaigrette and cornichons alongside.