Preheat oven to 450 degrees. In a small bowl, mix 1 tablespoon butter, 1 teaspoon thyme, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Rub evenly under chicken skin. Place on a rimmed baking sheet; roast skin side up until skin is golden brown and meat is cooked through, about 25 minutes.
Meanwhile, melt remaining tablespoon butter in a large saucepan over medium heat. Cook onion, stirring occasionally, until golden, about 8 minutes. Add sage and remaining 1/2 teaspoon thyme; cook 1 minute. Add broth; simmer until reduced to 1 1/2 cups, 10 to 15 minutes. Strain mixture; return to saucepan.
Add cranberries and sugar; boil until berries burst, 5 to 8 minutes. Whisk in cornstarch mixture; return to a boil. Cook until slightly thickened, 1 to 2 minutes. Remove from heat (sauce will thicken as it stands). Season with salt and pepper. To serve, spoon sauce over chicken.