Heat the water to boiling in a large pot. Add the bouillon cubes and dissolve.
Cut the onion into thin slices, then quarter the slices and add to the broth.
Add salt and pepper.
Bring the mixture back to a boil. Turn down the heat and simmer uncovered for 1 hour while stirring.
Sift the flour into the soup and continue to stir until smooth. Be careful when you stir so you do not tear the onions. As the soup continues to cook, any lumps should dissolve after 30 minutes of additional simmering.
Add the cream and 1 cup of cheddar cheese; continue to simmer the soup for another 5 to 10 minutes.
Serve the soup hot after sprinkling a tablespoon of each of the cheeses on top.