1. Put the grits in a medium pot, whisk in 5 cups cold water and season with salt. Bring to a simmer over medium-high heat. Cover and cook, stirring frequently and adding additional water as needed, until they are creamy and tender, 45 minutes to an hour or longer. Add butter to taste and season again with salt.
2. Meanwhile, prepare a hot fire in a charcoal grill.
3. Put the radicchio on a large plate and drizzle with olive oil, sprinkle all over with sugar and season generously with salt and pepper. When the flame has died down, the coals are completely covered with ash and the grill is very hot, grill the radicchio, starting with the cut side down until it is slightly charred on the outside and warm and starting to wilt on the interior, 2 to 3 minutes on each side.
4. Put the mozzarella in a bowl of hot, salted water for 30 seconds to heat and then drain.
5. Divide the grits among four warm plates, and arrange the cheese and radicchio alongside. Drizzle with balsamic and sprinkle with sea salt.
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