In a small saucepan over low heat, heat the olive oil with the butter and garlic until the butter is melted. Set aside and allow to steep for 5 minutes, then discard the garlic clove.
Heat the vegetable oil in a 3-quart Dutch oven over medium-high heat until the oil dances in the pan. Add the popcorn kernels, cover with a lid and cook, shaking the pan every couple minutes to prevent burning. When you begin to hear popping, lower the heat to medium and shake the pan every 30 seconds until there are very few pops left in the pan. Empty the popcorn into a large bowl; you should have 6 to 8 cups of popcorn. Toss with the garlic-butter mixture, the Parmesan and Pecorino, and the pepper and salt.