In a 5- to 6-quart pan over high heat, bring 1 quart water to a boil.
Peel and rinse potatoes; cut into 1-inch chunks. Add potatoes to boiling water; cover and return to a boil, then reduce heat to medium and simmer until potatoes mash easily, 10 to 12 minutes.
Meanwhile, in a 1- to 1 1/2-quart pan over medium heat, warm 1 cup milk and garlic until steaming, about 5 minutes. (Or in a 2-cup microwave-safe container, heat milk and garlic in a microwave oven on full power (100%) until steaming, 1 to 2 minutes.)
Drain potatoes and return to pan. Add the hot milk mixture, oil, cheese, basil, and paprika. Mash with a potato masher or an electric mixer until potatoes are as lumpy or smooth as you like, adding a little more milk for a softer, creamier texture if desired. Stir chives into mixture. Add salt and pepper to taste.