In a medium skillet, melt 1 tablespoon butter over medium-low heat. Add the onion and thyme, season with salt and pepper and cook until the onion is browned, about 20 minutes.
While the onion is cooking, in a large bowl, combine the sirloin and Worcestershire; season with salt and pepper. Form into 4 patties, making a depression at the center for even cooking. In a nonstick skillet, heat the EVOO, 1/2 turn of the pan, over medium-high heat until smoking. Add the patties and cook, turning once, for 6 to 8 minutes for medium. During the last minute of cooking, top the burgers with the cheese.
Toast the bread, then spread with the 2 tablespoons softened butter.
Stir the mustard and chicken stock into the onions. Stir in the parsley, then remove from the heat.
Place a beef patty on each of 4 buttered toasts; top with some of the onions and another buttered toast slice.