Char peppers on grill, under broiler, or over gas stovetop on high, rotating every 3 to 5 minutes until skin is completely blackened, about 15 to 20 minutes total. Once charred, move to a bowl, cover tightly with cling wrap, and set aside until peppers are cool enough to handle.

Fill a large pot with salted water and bring to a boil over high heat. Once boiling, add pasta and cook 1 to 2 minutes less than the recommended cooking time. Drain and set aside, rinsing with cold water to prevent sticking.

Warm olive oil in a large high-walled skillet or saucier over medium heat. Add pancetta and cook, stirring often, until fat is rendered and pancetta begins to brown, about 5 minutes. Add shallots and pinch of salt and sauté for 5 to 10 minutes, continuing to stir regularly.

Remove skins, stems, cores, and seeds from peppers, rinsing briefly to remove leftover skin. Add peppers and any liquid to a blender. Pulse several times, add cream, and puree until smooth, about 2 to 3 minutes, and set aside.

Once shallots are softened and translucent, add garlic, tomato paste, chile flakes, and paprika. Stir constantly, until fragrant and brick red, about 2 to 3 minutes. (If garlic begins to burn, reduce heat. If mixture is sticking, add a splash of olive oil.) Add vodka and cook until reduced by half and slightly thickened, about 3 to 5 minutes. Add red pepper mixture and stir. Add 1 teaspoon of salt, stir, and adjust as needed, adding 1⁄4 teaspoon at a time. Add pasta and cook, about 1 to 2 minutes, stirring to thoroughly coat. Serve topped with burrata or parmesan, flaky sea salt, and basil.