Preheat the oven to 400°F. Line a rimmed baking sheet with foil.

In a large skillet, melt the butter with 2 tablespoons olive oil over low heat. Stir in half of the garlic, 1 teaspoon each basil and oregano and ¼ teaspoon red pepper flakes. Scrape the butter mixture into a small bowl. (No need to wipe out the skillet.)

Using a serrated knife, cut the whole pack of rolls in half horizontally, keeping the rolls attached. Place the top and bottom halves cut-side up on the baking sheet. Brush the cut sides with about 4 tablespoons of the melted butter mixture. Toast in the oven until crisp and light golden, about 5 minutes. Set aside; leave the oven on.

Meanwhile, heat the remaining 2 tablespoons olive oil in the skillet over medium heat. Add the turkey and season with the salt. Cook, breaking up the meat with a wooden spoon, until no longer pink, 4 to 5 minutes. Add the onion and cook until it begins to soften, about 4 minutes.

Make a space in the middle of the pan and add the tomato paste, remaining garlic, and remaining 1 teaspoon each basil and oregano and ¼ teaspoon red pepper flakes. Let sizzle for a minute, then stir into the turkey. Add ½ cup water and stir. Cook until the pan is almost dry, 1 to 2 minutes.

Spoon the turkey mixture over the bottom half of the rolls, spreading it to the edges. Press the meat down lightly to keep it in place. Sprinkle with the mozzarella and 2 tablespoons parmesan.

Spread the cut side of the roll tops with the pesto, then flip over onto the sliders. Brush the tops of the rolls with the remaining melted butter mixture and sprinkle with the remaining 2 tablespoons parmesan.

Bake until the cheese is melted and the rolls are crisp and lightly browned on top, about 10 minutes.