Add olive oil to the skillet. Once it gets shimmery, add the onions and garlic and sauté until creamy, about 3 minutes.
Add the finely diced kimchi and cook for 3 minutes.
Add the browned turkey sausage back in along with the diced tomatoes and mix. Season with salt and pepper and simmer on medium-low until the tomato sauce has thickened, around 10 minutes.
Turn off the heat. Stir in the crumbled feta cheese and then crack the eggs into the skillet, trying to space them out evenly.
Bake 7 to 10 minutes and serve with more feta, cilantro, and bread.