Microwave vinegar, ⅛ teaspoon salt, and ⅛ teaspoon cayenne in medium bowl until steaming, 1 to 2 minutes. Add radishes and let sit, stirring occasionally, for 45 minutes. Drain radishes and return to now-empty bowl; set aside for serving.
Pulse mushrooms in food processor until finely chopped, about 25 pulses, scraping down sides of bowl as needed. Heat 2 tablespoons oil in 12-inch nonstick skillet over medium heat until shimmering. Add mushrooms and remaining ¼ teaspoon salt, cover, and cook until mushrooms have released their liquid, about 5 minutes. Uncover and continue to cook, stirring occasionally, until mushrooms are lightly browned and dry, about 10 minutes; transfer to large bowl.
Process cilantro, parsley, coriander, cumin, garlic, and remaining ¼ teaspoon cayenne in now-empty processor until finely ground, about 30 seconds, scraping down sides of bowl as needed. Stir herb mixture, flour, and egg white into mushrooms until combined. Cover and let sit at room temperature for 15 minutes.
Pinch off and roll mushroom mixture into 1½ tablespoon-size balls (you should have 12 balls). Flatten balls into ¾-inch-thick patties.
Heat remaining 3 tablespoons oil in now-empty skillet over medium heat until shimmering. Cook patties until deep golden brown, 3 to 5 minutes per side. Transfer patties to paper towel–lined plate and let drain briefly. Divide patties evenly among lettuce leaves and top with cucumber, radishes, and tahini sauce. Serve.