Combine yogurt, garlic and salt in a medium bowl, mixing just until combined. Divide yogurt sauce evenly among 2 serving plates or shallow bowls, spreading with a small spatula or back of a spoon to make a flat bed with a small indent large enough to hold 1 poached egg. Set aside, allowing the yogurt to warm up while you poach the eggs.

Fill an 8- inch skillet with water up to the rim. Add vinegar and salt bringing the water to a rolling broil. This helps the egg whites set up with fewer wispy or floating white pieces in the skillet. Meanwhile, crack the eggs, one at a time into a colander set over a bowl. Let stand until loose, watery parts drain away from eggs, about 30 seconds. Gently transfer eggs to a small teacup or bowl.

With the edge of the teacup just above surface of water, gently tip eggs into the water. Cover the skillet, remove it from the heat and let the eggs gently steam until the whites are set but the yolks are still slightly runny, about 3 minutes. If the whites closest to the yolk is not set, let stand in water, checking every 30 seconds, until eggs reach desired doneness.

While the eggs cook, heat butter in a microwave until it begins to sputter, about 30 seconds. Remove from microwave, stir in pepper flakes.

Using a slotted spoon, carefully lift and drain 1 egg over pot. Pat bottom of spoon dry with paper towels. Now your egg is ready to be placed on the garlicky yogurt bed and drizzled with spicy butter and a pinch of dried mint. Serve immediately, passing toasted slices of rustic bread separately.