Using 3‐inch (8 cm) round cutter, cut out circle from centre of each bread slice; reserve centres.
In large, nonstick skillet, heat butter and oil over medium heat; cook bread slices until lightly toasted, approximately 1 to 2 minutes. Flip and crack 1 egg into hole of each bread slice.
Season eggs with salt and pepper. Cover and cook until whites are set and yolks are cooked to desired doneness, approximately 2 to 4 minutes. Transfer to plates.
Toast reserved bread pieces for 1 minute per side or until golden brown. Top eggs with salsa and feta cheese. Serve with toasted bread rounds.