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Stir yogurt, garlic, and salt in medium bowl until just combined. Divide yogurt mixture evenly among 4 serving plates or shallow bowls, spreading each portion with small spatula or back of spoon to make flat bed large enough to hold 1 poached egg. Set aside plates and allow yogurt to warm up while you prepare eggs.
Bring 1½ quarts water to boil in Dutch oven over high heat. Meanwhile, crack eggs, one at a time, into colander. Let stand until loose, watery parts of whites drain away from eggs, 20 to 30 seconds. Gently transfer eggs to 2-cup liquid measuring cup.
Add vinegar and 1 teaspoon salt to boiling water. Remove pot from heat. With lip of measuring cup just above surface of water, gently tip eggs into water, one at a time, leaving space between them. Cover pot and let stand until whites closest to yolks are just set and opaque, about 3 minutes. If after 3 minutes whites are not set, let stand in water, checking every 30 seconds, until eggs reach desired doneness.
While eggs cook, heat butter in small saucepan over medium heat until it sputters, 2 to 3 minutes. Stir in pepper flakes (butter will foam) and remove from heat.
Using slotted spoon, carefully lift and drain 1 egg over pot. Pat bottom of spoon dry with paper towel and gently place egg on yogurt bed. Repeat with remaining eggs. Drizzle butter evenly over eggs. Sprinkle each serving with pinch dried mint, if using, and season with salt and pepper to taste. Serve immediately, passing pita separately.