1. Heat oven to 425°F. Heat butter and oil in a 6-qt. saucepan over medium-high heat; add garlic and onion, and cook, stirring often, until soft, about 6 minutes.
2. Stir in flour, and cook until smooth, about 1 minute. Add milk and cream, and stir until smooth. Add potatoes, rutabagas, and 2 teaspoons thyme, and bring mixture to a boil; cook, stirring often, until vegetables are slightly tender and broken apart, about 5 minutes.
3. Add in half the cheese and salt and pepper, and then transfer to a 9? × 13? baking dish; top with remaining cheese and bake until golden brown and bubbling, about 25 minutes. Sprinkle with remaining thyme before serving.