In a large saucepan, bring the chicken broth, half-and-half, butter and 1/2 teaspoon salt to a boil. Add the grits, reduce the heat and simmer, whisking, until thick, 10 to 15 minutes. Stir in the cheese, chili and adobo sauce and heat through, 2 to 3 minutes. Remove from the heat and cover to keep warm.
In a medium skillet, heat 2 teaspoons oil over medium heat. Fry 2 eggs; season with salt and pepper and serve over some of the grits with extra adobo sauce, scallions and cilantro, if using. Repeat with the remaining oil and eggs.
