Preheat oven to 250 degrees. Cook rice according to package instructions; cover, and set aside. In a wide, shallow bowl, beat eggs; in another, combine 1 cup flour with paprika. Season eggs, flour mixture, and steaks with salt and pepper. Dredge each steak in flour, then dip in egg; allow excess to drip off, then dredge in flour again.
In a large cast-iron or nonstick skillet, heat 1/4 cup oil over medium-high. Cook steaks in skillet until crust is golden, 3 to 4 minutes per side. Transfer to a baking sheet, and place in oven to keep warm.
Pour off all but 1 tablespoon oil from skillet. Add remaining teaspoon flour; cook, whisking until golden, about 30 seconds. Gradually add milk while whisking; raise heat to high, and boil until thickened, 4 to 5 minutes. Season gravy with salt and pepper. Cover, and set aside.
In a medium skillet, heat remaining teaspoon oil over medium-high. Add onion; cook, stirring occasionally, until softened, about 4 minutes. Add green beans and 1/3 cup water; cover, and cook until green beans are bright green, 1 to 2 minutes. Uncover; continue to cook until green beans are crisp-tender and onion begins to brown, 3 to 4 minutes more. Reheat gravy, thinning with milk if necessary. Serve steak with green beans, rice, and gravy.