Gently mix the ground beef, 1 tablespoon teriyaki sauce, 3/4 teaspoon salt and 1/2 teaspoon pepper in a large bowl. Form into four 4 1/2-inch patties. Melt 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the patties and cook until browned, about 3 minutes per side for medium. Transfer to a platter.
Add the mushrooms to the skillet and cook until browned, about 3 minutes. Sprinkle in the flour and stir until lightly toasted, about 1 minute. Whisk in the beef broth, ginger and remaining 2 teaspoons teriyaki sauce. Simmer over medium-high heat until reduced to a thick gravy, 3 to 5 minutes.
Meanwhile, heat the remaining 1 tablespoon butter in another large nonstick skillet over medium-high heat. Crack the eggs into the skillet, reduce the heat to medium and partially cover. Cook until the whites are set but the yolks are still runny, about 3 minutes. Season with salt and pepper.
Serve the burgers on the buns; top with the mushroom gravy and fried eggs; drizzle with Sriracha. Serve with pasta salad and pickles.