For the grits: Bring the milk, salt and pepper to a boil in a heavy-bottomed pot. As soon as the milk comes to a boil, gradually whisk in the grits. Turn down the heat to a simmer and slowly cook the grits, stirring to keep from sticking, until thick and creamy, 15 to 20 minutes.
Remove the grits from the fire and stir in the butter and 1/2 cup of the green chili puree. Fold in the cheese and green onions and serve immediately.