Preheat the oven to 500 degrees F.
Brush 1 tablespoon of oil over each of 2 heavy large lined baking sheets. Place the buttermilk in a large bowl. Add the chicken tenders and stir to coat. Let stand at least 15 minutes and up to 30 minutes.
Meanwhile, mash the garlic with the salt in a medium bowl. Whisk in the vinegar and then the remaining 1/2 cup of oil. Season the vinaigrette, to taste, with pepper. Transfer the vinaigrette to a small serving bowl.
Stir the Parmesan and bread crumbs in a pie dish. Remove the chicken tenders from the buttermilk and dredge them in the bread crumb mixture to coat completely, pressing to adhere. Arrange the coated chicken tenders on the prepared baking sheets, spacing evenly. Drizzle the remaining 2 tablespoons of oil over the chicken tenders and bake until they are cooked through and golden brown, about 12 minutes.
Transfer the chicken tenders to a platter and serve the vinaigrette alongside for dipping.