Vietnamese Banh Mi Chicken Burger

Name: Vietnamese Banh Mi Chicken Burger
Ingredients: 1 small cucumber, thinly sliced; 1 medium carrot, peeled and sliced into ribbons; 3 radishes, thinly sliced; 1 cup white vinegar; 2 1/2 teaspoon salt, divided; 1 teaspoon sugar; 6 ounces firm tofu; 1 egg; 1/4 cup scallions, chopped; 1/2 teaspoon black pepper, divided; 1 1/4 pounds ground chicken; 1 tablespoon olive oil; 2 teaspoons sesame oil; 1/2 pound shiitake mushrooms; 2 teaspoons soy sauce; 1 1/2 teaspoon apple cider vinegar; 1/4 teaspoon ground cumin; 1/8 teaspoon brown sugar; 1 cup bean sprouts; 1 bunch cilantro, chopped; 4 whole-wheat hamburger buns