Preheat the oven to 400°F. Wrap the sweet potato in foil, place on a baking sheet, and bake until tender, 35 to 45 minutes. Let cool completely.

In a large saucepan, bring the broth and milk to a gentle simmer over medium heat.

Place the grits in a bowl, cover with water, and swish to loosen the chaff. Drain through a fine-mesh sieve. Repeat rinsing and draining two more times.

Whisk the rinsed grits into the simmering liquid. Cook over medium heat, whisking often, until thick and creamy, about 25 to 30 minutes.

Unwrap, peel, and mash the sweet potato. Stir the mashed potato into the cooked grits. Blend with an immersion blender until smooth.

Stir in the butter and season with salt and pepper. Transfer to a serving bowl and top with a pat of butter and freshly ground black pepper.