In a large saucepan, combine the water, milk, onion, garlic, olive oil, salt, and bring to a boil.
Reduce heat and simmer over moderate heat until the onion and garlic are softened, about 3 minutes. Return to a boil and whisk in the grits. Cook over low heat, stirring frequently with a wooden spoon, until the grits are tender, about 25 minutes.
Add the cheese, scallions, and thyme and stir until the cheese has melted. Transfer to bowls and serve.