Heat a wide skillet over medium heat. Add the butter and olive oil and cook until the butter melts and foams quietly.
Add the garlic and thyme and cook until fragrant and pale golden, about 1 minute, adjusting the heat to prevent browning.
Crack the eggs directly into the skillet, spacing them evenly. Season with salt and black pepper.
Scatter the feta evenly over the eggs and butter. With the tip of a knife, lightly pierce each yolk to loosen it slightly without fully breaking.
Cover the skillet tightly with foil and cook until the whites are just set and the yolks remain soft, 3 to 4 minutes.
Remove from heat and serve immediately, with warm, crisp bread for scooping.