Heat 3 tablespoons of the oil in a large skillet over medium heat.
Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper. Dredge in the flour, then in the eggs (letting any excess drip off), then in the bread crumbs.
Working in two batches (using the remaining 3 tablespoons oil for second batch), cook the chicken until golden brown, 2 to 3 minutes per side. Transfer to a paper towel-lined plate.
Bring a pot of salted water to a boil. Add the green beans. Cook until tender, 3 to 5 minutes. Drain and return to pot. Add the butter and ¼ teaspoon salt.
Divide the chicken among plates and serve with green beans.