Combine 1/2 cup water, the vinegar, sugar and 3/4 teaspoon salt in a small saucepan. Cook over medium heat, stirring, until the sugar dissolves, about 2 minutes. Combine the cucumbers and carrots in a bowl; pour in the hot liquid and refrigerate until ready to serve.
Combine the ground beef and 1 tablespoon Worcestershire sauce in a large bowl. Mix with your hands until just combined. Form into four 1-inch-thick patties; season with salt and pepper. Set aside.
Melt 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the mushrooms, season with salt and cook, stirring occasionally, until browned around the edges, about 10 minutes. Transfer to a plate and cover to keep warm.
Wipe out the skillet and melt the remaining 1 tablespoon butter. Add the burgers and cook until browned, about 4 minutes per side; top with the cheese and cover during the last 1 minute.
Meanwhile, mix the ketchup with the remaining 1 tablespoon Worcestershire sauce; spread on the bottom buns. Serve the burgers on the buns; top with the mushrooms. Drain the pickles and serve on the side.