Season beef with salt and pepper and shape into eight patties about 2 inches wide and 1/2 inch thick. Heat broth and water together in a saucepan and keep at a bare simmer.
Heat oil in a 12-inch deep skillet over medium-high heat. Cook patties for 3 to 4 minutes per side until browned and 160°F in the centers; transfer to a plate.
Leave 1 tablespoon fat in the skillet, lower heat to medium and cook onion for 4 minutes. Stir in dry arborio rice for 1 minute.
Add 1 cup hot broth and stir, scraping the browned bits from the skillet. Simmer gently, adding more broth about 1/2 cup at a time as the previous addition is nearly absorbed and stirring every minute or two.
Continue for 22 to 28 minutes until rice is tender with a slight bite and flows slowly when stirred; use as much hot broth as needed. Stir in ketchup and mustard.
Return patties and their juices to the rice for 2 minutes to warm. Remove from heat, gently stir butter and cheddar through the rice around the patties and rest for 2 minutes before serving two patties per person.