Preheat oven to 400 degrees Fahrenheit.
In a medium bowl, whisk together eggs, milk, lemon zest, lemon juice, salt, and pepper.
In a 10 inch oven-safe skillet, drizzle the olive oil and heat to medium low. Chop the cooked spaghetti into about 3 inch pieces and evenly distribute across heated skillet. Cover the spaghetti with spinach. Add the whisked eggs and gently stir to even out all the ingredients. Cook until the egg sets around the edges, but is still slightly jiggly in the middle.
Transfer skillet to preheated oven and bake until cooked through, 8-10 minutes.
Top with feta cheese, cut into slices, and garnish with basil.