How to Make It
Preheat oven to 350°F. Mix yogurt, dill, garlic, ¼￼ teaspoon kosher salt, and several grinds of pepper in a small bowl.
Melt butter in a small skillet over medium-low. Add sesame seeds, red pepper flakes, and a pinch of flaky salt; cook until fragrant, about 2 minutes. Remove from heat. Place pitas on a baking sheet, brush both sides with oil, and season with a pinch each kosher salt and pepper. Bake until puffed and warm, about 5 minutes.
Meanwhile, bring 2 inches of water to a gentle simmer in a large skillet. Crack 4 eggs into skillet and poach until whites are set but yolks are still runny, about 3 minutes. Using a slotted spoon, carefully transfer cooked eggs to paper towels to drain. Repeat with remaining 4 eggs.
Arrange pitas on serving plates. Top with yogurt, eggs, chile butter, dill sprigs, mint, and a pinch of flaky salt.