Combine all of the ingredients in a small mixing bowl. Whisk until well combined. Refrigerate until ready to serve. Heat the olive oil in a medium saucepan over medium heat. Add the mushrooms and cook until tender, about 5 minutes. Strain the mushrooms and let cool. Place the cooked mushrooms, beef, salt, and pepper in a mixing bowl. Mix to combine. Form the beef mixture into 4 patties of equal size. Heat a grill or grill pan over high heat. Cook the burger patties to desired doneness, about 5 minutes per side for medium. Sandwich each patty with lettuce and pickles between a bun and serve with Ginger-Lime Mayonnaise on the side.