Turkey Shawarma With Crunchy Vegetables

Name: Turkey Shawarma With Crunchy Vegetables
Ingredients: 1 cup plain whole-milk Greek yogurt; 1 garlic clove, finely grated; 1 teaspoon finely grated lemon zest; 3 teaspoons fresh lemon juice, divided; Kosher salt; 1/2 teaspoon freshly ground black pepper; 1/2 teaspoon ground coriander; 1/2 teaspoon ground cumin; 1/2 teaspoon smoked paprika; 1/4 teaspoon ground cinnamon; 1/4 teaspoon ground turmeric; Pinch of cayenne pepper; 4 tablespoons unsalted butter, divided; 2 tablespoons vegetable oil, divided; 4 cups shredded turkey meat, room temperature, divided; 1 small red onion, halved, very thinly sliced; 2 medium Persian cucumbers, cut into matchsticks; 1 medium carrot, peeled, cut into matchsticks; 4 radishes, trimmed, thinly sliced; 1 tablespoon plus 2 teaspoons red wine vinegar; 2 cups mixed tender herbs (such as torn mint leaves, parsley leaves with tender stems, and/or dill fronds); 6 white pitas with pockets, warmed; Hot sauce (for serving)