Adjust oven rack to upper-middle position and heat oven to 425 degrees. Toss cheddar and 1 teaspoon cornstarch together in bowl until well combined. Toss Parmesan and remaining teaspoon cornstarch together in second bowl until well combined.

Heat oil in 8-inch nonstick skillet over medium heat until simmering. Add onion and cook until softened and lightly browned, 5 to 7 minutes. Stir in garlic and thyme and cook until fragrant, about 30 seconds. Stir in chicken broth, cream, potatoes, salt, and pepper and bring to boil. Reduce heat to medium, cover, and simmer until potatoes are nearly tender, 10 to 12 minutes.

Off heat, stir in cheddar mixture and press potatoes into even layer, removing any air pockets. Sprinkle Parmesan mixture evenly over top and bake until golden brown, 12 to 14 minutes. Transfer potatoes to wire rack and let cool for 5 to 10 minutes before serving.