Mix the strained yogurt with the grated garlic in a bowl. Set aside.
In a medium skillet, heat the olive oil with the chili flakes over medium-high heat until it ripples, about 2–2½ minutes.
Crack the eggs in gently, holding the shell close to the oil. Let them cook untouched until the edges are golden, 2–3 minutes.
Tilt the pan and baste the whites and yolk edges with the hot oil until the whites are set, about 1 minute — keep oil off the yolk itself. Season with salt and pepper.
Spread the garlic yogurt on two plates, top each with a fried egg, spoon over remaining chili oil, scatter with dill, and serve immediately with toasted bread.
