Place rinsed black beans in a medium bowl; mash with a fork until mostly smooth with some pieces remaining.
Add ⅔ cup mushrooms, ½ cup onion, ½ cup panko, the beaten egg, 1 tablespoon chipotle, the minced garlic and ½ teaspoon salt; stir until well combined.
Divide the mixture into 4 (about 4-ounce) portions and shape into firm balls. Cover and refrigerate for at least 30 minutes or up to 12 hours.
Combine ¼ cup yogurt, 2 teaspoons ketchup and ½ teaspoon mustard in a small bowl.
Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add 2 bean balls to the skillet.
Coat the bottom of a sturdy spatula with cooking spray and firmly press each ball into a 4-inch patty.
Cook, undisturbed, until well browned and crisp around the edges, 2 to 3 minutes.
Flip the patties and top each with 1 cheese slice. Cover and cook until the cheese is melted and the patties are heated through, about 2 minutes.
Transfer to a plate. Repeat the process with the remaining 1 tablespoon oil, 2 bean balls and 2 cheese slices.
Spread about 1½ teaspoons sauce on each toasted bun half.
Layer the bottom halves with a bean patty, 1 lettuce leaf and 1 tomato slice. Top with the top bun half.
