Heat oil in a medium skillet over medium-high heat. Add onion and cook, stirring, until softened and beginning to brown, about 2 minutes. Add spinach and cook, stirring, until wilted, 1 to 2 minutes more. Divide the spinach mixture among the pepper halves. Sprinkle with feta, then top with the egg mixture. Bake until the filling is set, 30 to 35 minutes.