Heat oven to 400º F. Toss the tomatoes with 1 tablespoon oil and ¼ teaspoon salt. Arrange, cut-side up, on a baking sheet. Bake until just softened, about 10 minutes.
Meanwhile, whisk together the Feta, yogurt, lemon juice, and ¼ teaspoon of the salt. Set aside.
Heat ½ tablespoon oil in a medium skillet over medium heat. Add the garlic and red pepper and cook, stirring occasionally, until the garlic is softened and golden, 1 to 2 minutes.
Add the spinach and ½ teaspoon salt. Cook, tossing occasionally, until the spinach is wilted, about 4 minutes.
Brush both sides of the pitas with the remaining ½ tablespoon oil. Place in oven and heat until lightly toasted, about 2 minutes per side. (You can also do this on a medium-hot grill.)
Place the pitas on plates. Top with the spinach and tomatoes. Drizzle the yogurt sauce on top.