Heat 1 tablespoon olive oil in a 4 to 6-inch nonstick skillet. In a food processor with grating attachment, grate potato. Transfer potato to skillet. Pat into thin pancake. Cook until golden on one side, 4 to 5 minutes. Turn pancake over and cook until golden on other side, 4 to 5 minutes more.
Pour 3 tablespoons olive oil into another non-stick skillet. Thinly slice 2 cloves of garlic. Place garlic in cold oil and bring up to heat slowly. Saute until garlic is just golden. Remove garlic from pan and add spinach. Cook until just wilted. Remove from pan.
Beat 4 eggs until just combined. Do not beat in too much air into eggs or the texture will be dry. Add 1 tablespoon butter to spinach pan. After foam has subsided, pour in the eggs. While stirring the eggs with a fork, shake the pan back and forth.
Once the bottom of the omelet has set, slide in potato pancake. Add spinach and crumbled feta cheese on top of potato. Run omelet under broiler until cheese starts to melt. Remove from broiler and fold into half moon shape. Sprinkle sauted garlic slices over top of omelet.