If you haven't yet, cook the rice. Set aside and allow it to cool.

Make a roux by melting the butter over medium heat, and then adding the flour. Whisk to combine. Then slowly add the milk, a little at a time, whisking everything together in between additions of milk. This will form a thick paste, and eventually a thick sauce.

Combine the rice, egg yolks, salt, parmesan, and roux. If it's still too warm to handle, you can mix it and then spread thinly into a pan. Refrigerate until cool.

Shape the mixture into croquettes: they can look like small, 2" wide hockey pucks or 3" long corndog-like cylinders. Then dredge the croquettes first in the egg whites and then in the breadcrumbs.

Heat your oil sufficiently over medium-high heat, and fry the croquettes until golden brown, 1-2 minutes on each side. Drain them on a paper towel, and then add them to your soup.