Form beef into four patties; season with salt and pepper. In a large skillet, heat oil over medium. Add patties; cook 6 to 8 minutes per side for medium. Transfer to a plate, and tent with aluminum foil; set aside.
To skillet, add green beans, onion, and cup water; season with salt and pepper. Cover, and cook, tossing occasionally, until beans are crisp-tender, 5 to 7 minutes. Uncover, and cook, tossing, until water has evaporated, about 2 minutes.
Make sauce: In a small bowl, stir together mayonnaise, parsley, and horseradish; season with salt and pepper. Place tomato slices and burgers on toast; top with horseradish sauce, and serve with green beans.
Tangy horseradish sauce is a piquant alternative to ketchup or mustard. We used parsley in the sauce, but any fresh leafy herb will taste equally good.