In a small bowl, combine the room-temperature yogurt, grated garlic, and a pinch of salt. Mix until smooth and divide between two shallow bowls.
In a small skillet or saucepan, melt the butter over medium heat until it starts to foam. Add the Aleppo pepper, swirl for 30 seconds until the butter turns bright red, then remove from heat.
Bring a pot of water to a gentle simmer (not a rolling boil) and add the vinegar.
Crack an egg into a small ramekin, create a gentle whirlpool in the water, and drop the egg into the center. Poach for 3 minutes. Repeat with the second egg.
Use a slotted spoon to remove the eggs and pat them dry on a paper towel.
Place the poached eggs on top of the yogurt.
Drizzle the warm chili butter over the eggs and yogurt.
Top with fresh dill, mint, and an extra pinch of salt. Serve immediately with toasted bread.