For the brown butter:
6 tablespoons (3/4 stick) unsalted butter
1 sprig thyme
For the grits:
2 tablespoons unsalted butter
1 medium onion, very finely chopped (3/4 to 1 cup)
2 1/2 cups water
3 1/2 cups whole milk, or more as needed
1/4 teaspoon kosher salt
1/4 teaspoon freshly ground black pepper, or more to taste
3/4 cup stone-ground grits
15 ounces (1 can) pumpkin puree (may substitute sweet potato puree)
Leaves from 1/2 bunch thyme, chopped (3 tablespoons; may substitute leaves from 2 stems of sage)
1/2 cup freshly grated Parmigiano-Reggiano cheese
For the brown butter: Melt the butter in a medium skillet over medium heat, then cook for 3 to 4 minutes, stirring occasionally, until the butter just turns brown. Transfer to a heatproof bowl and add a sprig of thyme, making sure it is completely submerged.
For the grits: Melt the butter over medium-low heat in the same skillet used to cook the brown butter. Add the onion; cook for about 8 minutes, stirring occasionally, until softened and almost creamy. Remove from the heat; transfer the onion and butter to a bowl.
Combine the water, 3 cups of the milk, salt and pepper in a large, heavy-bottomed saucepan over high heat; bring to a boil. Use a long-handled whisk to slowly mix in the grits. Be careful; the grits will soon begin to sputter. As soon as the mixture has thickened a bit, reduce the heat to medium or medium-low and cook for 30 to 35 minutes, whisking often, scraping the sides and bottom of the saucepan and adjusting the heat as needed to prevent scorching, until the grits have thickened to the consistency of porridge. Add milk if the grits become too thick.
Add the onion, pumpkin puree and thyme leaves to the grits; cook for 3 minutes, stirring or whisking continuously. Remove from the heat; add the remaining 1/2 cup of milk and the Parmigiano-Reggiano; stir until the cheese is thoroughly incorporated. Taste, and add pepper as needed.
Divide among individual bowls. Drizzle each portion with some of the brown butter (discarding the thyme sprig). Serve hot.