Allow the steaks to come to room temperature, then rub each steak with 1 tablespoon of the olive oil and season each steak with 2 teaspoons of the kosher salt and 1 teaspoon of the cracked black pepper.

Preheat a grill to medium-high. Grill the steaks, rotating each steak 90 degrees midway through cooking on each side, about 6 to 8 minutes per side for medium rare. Transfer the steaks to a platter to rest for 5 minutes before serving.

When ready to serve, slice the 2 sides of each steak away from the t-shaped bones and slice both portions of each steak into very thin slices against the grain of the meat. Divide the arugula evenly among 6 large dinner plates and divide the steak slices evenly among the top of the greens on each plate. Drizzle any accumulated meat juices over the top of the meat. Sprinkle 2 teaspoons of the lemon juice and 1 tablespoon of the remaining olive oil over each plate. Sprinkle 1/4 teaspoon of the sea salt over each plate and garnish with strips of Parmigiano-Reggiano, if desired. Serve immediately with the radicchio and onions.

Preheat the grill to medium-high.

Cut the radicchio heads in half lengthwise and brush with olive oil. Sprinkle with salt and pepper. Place radicchio on the grill, cut side down, and grill until lightly charred. Remove from heat and serve with steaks.

Fill a skillet or saucepan with a tight-fitting lid with 1 or 2 inches of water. Bring to a simmer, add onions, and cook until tender. Remove lid and add sugar and balsamic. Continue cooking until vinegar mixture is reduced to a glaze. Serve with the steaks.