Combine 2 tbsp. salt and the sugar in a small bowl. Rub both sides of tuna belly with salt mixture. Set on a plate, cover, and chill at least 6 hours and up to overnight.
Briefly rinse fish with cold water and pat dry with paper towels. Using a very sharp knife, thinly slice belly diagonally across the grain, pulling the knife through the fish rather than sawing. Divide slices among 4 to 6 small salad plates, spreading them out in a single layer. Cover each with plastic wrap and, using the bottom of a liquid measuring cup, gently pound fish into a thin layer.
Thinly slice radishes, then cut into thin matchsticks. Transfer to a small bowl and add avocado, chives, oil, and lemon juice. Season with remaining 1/2 tsp. salt and toss until well combined. Spoon some salad onto each plate.