Put all brine ingredients except for ice in a large pot to boil. Cook for ten minutes then remove from flame. Add ice and allow to cook. Add beef cheeks and allow to sit overnight.

Remove beef cheeks from brine and season with salt and pepper. Heat oil in a heavy pot or Dutch oven and sear the cheeks on all sides. Put the cheeks aside.

Add onions, carrots, and celery to the pot and caramelize.

Add the tomato paste and stir until also caramelized, and add the wine and cook until reduced by about half. Add the beef stock and the cheeks and bring to a boil. Cover the pot with aluminum foil and place in oven set at 275 degrees Fahrenheit. Cook for 4 hours or until tender.

Roast potatoes and then peel. Rice potatoes, then fold in egg yolks, flour, and salt. Mix well to incorporate. Let rest for 30 minutes.

Roll out to a half-inch (one- and- a- half-centimeter) thickness and cut into half-inch (one- and- a- half-centimeter) squares. Bring a pot of salted water to a boil and gently place gnocchi into the boiling water. Let cook for four to five minutes after they float to the top and then check for doneness.

Cut the beet, rutabaga, turnip, and carrot into different shapes and then poach in olive oil separately.

Sauté the gnocchi in oil until golden. Add the rosemary and poached vegetables and heat until hot.

Plate with the cheeks and reduce the braising liquid for the sauce.