Scoop the yogurt into a paper coffee filter set in a strainer over a bowl and let drain in the refrigerator for 2 hours. Transfer the yogurt to a bowl and discard the liquid.
In a small skillet, heat the olive oil. Add the garlic and cook over moderate heat until fragrant and lightly golden, about 2 minutes. Let the garlic oil cool slightly, then stir into the yogurt and season with salt and cayenne.