Heat a large skillet over medium-high heat. Season the steaks with salt and pepper. When the pan is hot, add 1 tablespoon butter, swirling the pan to coat. Add the steaks, sprinkle them with a pinch of sugar, and sear until golden brown on the bottom, about 4 minutes. Flip and cook until golden on the other side, 2 more minutes. Transfer to a cutting board and tent with foil.
Reduce the heat to medium and add 1 tablespoon butter to the skillet. Add the garlic and cook 20 seconds. Add the radicchio in batches, tossing until wilted. Sprinkle with 1 to 2 teaspoons sugar and 1/2 teaspoon salt. Add the sun-dried tomatoes and vinegar and cook until the vinegar evaporates slightly, about 1 minute. Add the broth and cook until the radicchio is tender, about 2 minutes. Stir in the basil. Transfer the radicchio to plates using tongs. Continue cooking the sauce until it thickens slightly, 1 to 2 minutes. Stir in the remaining 1 tablespoon butter.
Thinly slice the steak and add to the plates. Drizzle with the sauce and top with more sun-dried tomatoes and basil.