Preparation Generously cover potatoes and 1 tablespoon salt with cold water in a medium pot, then simmer until tender, about 12 minutes. Drain potatoes and rinse with cold water. Cool to room temperature in refrigerator, about 40 minutes. Meanwhile, cook garlic in oil at a bare simmer in a small heavy saucepan, stirring occasionally, until tender and golden, about 12 minutes. Drain garlic, reserving oil for another use, then mash to a paste. Preheat oven to 350°F with rack in middle. Mince basil, then stir together with garlic paste, cheese, mayonnaise, and salt and pepper to taste. Using a small melon-ball cutter or a teaspoon, scoop out a hole in each cooled potato, leaving a 1/4-inch shell. Stuff potatoes with cheese mixture and bake in a pan until cheese is melted and bubbling, about 20 minutes.
Preparation Generously cover potatoes and 1 tablespoon salt with cold water in a medium pot, then simmer until tender, about 12 minutes. Drain potatoes and rinse with cold water. Cool to room temperature in refrigerator, about 40 minutes. Meanwhile, cook garlic in oil at a bare simmer in a small heavy saucepan, stirring occasionally, until tender and golden, about 12 minutes. Drain garlic, reserving oil for another use, then mash to a paste. Preheat oven to 350°F with rack in middle. Mince basil, then stir together with garlic paste, cheese, mayonnaise, and salt and pepper to taste. Using a small melon-ball cutter or a teaspoon, scoop out a hole in each cooled potato, leaving a 1/4-inch shell. Stuff potatoes with cheese mixture and bake in a pan until cheese is melted and bubbling, about 20 minutes.