Preheat the oven to 400 degrees F.
Cut off the top 1/4-inch of the head of garlic, rub with the olive oil and wrap loosely with aluminum foil. Place on a small baking sheet and bake until tender and golden, 30 to 40 minutes. Allow the garlic to cool, then squeeze the cloves from their skins into a medium bowl and mash with a fork until smooth. Add the mayonnaise, 1/4 teaspoon of the paprika, the lemon juice and 1/4 teaspoon of the cayenne. Set aside.
While the garlic is roasting, make the spice blend. In a small mixing bowl, stir together the remaining 1 teaspoon paprika and 1/4 teaspoon cayenne, the garlic powder, onion powder, oregano, thyme, salt and pepper. Mix in the Pecorino Romano. Set aside.
Heat 2 inches of vegetable oil to 350 degrees F in a Dutch oven or heavy bottomed saucepot, or set a countertop fryer to 350 degrees F. Working in 2 batches, cook the fries until golden and crisp, about 4 minutes per batch. Drain the fries, using a large slotted spoon, and put into a large bowl. While the first batch is hot, toss with half of the spice mix until well coated. Repeat with the second batch of fries and the remaining spice mix. Transfer to a serving bucket and serve hot with the garlic mayo on the side.