Put the turkey cutlets in a glass bowl and add the garlic-flavored olive oil and vinegar and season with salt and pepper, to taste. Toss to coat, and let stand for 10 minutes.
Preheat a grill pan over medium-high heat. Bring a large pot of water to a boil, over medium heat, and add a generous amount of salt.
Remove the turkey from the marinade, shaking off any excess liquid and transfer to a plate. Drizzle with oil to keep it from sticking and season well with salt and pepper. Grill the cutlets, turning once, until the turkey is cooked through, about 1 1/2 minutes per side. Transfer to a cutting board and chop into bite-sized pieces.
Boil the pasta according to the package instructions. Add the asparagus in the last 2 minutes of cooking time. Carefully remove about 1/2 cup of the cooking water and reserve. Drain the pasta in a colander, add some garlic oil to the bottom of the pot and return the pasta to the pot, off the heat. Add the turkey and any juices released to the cooked pasta and toss to combine.
Season the pasta with salt and pepper, to taste and drizzle with additional olive oil and vinegar, if desired. Add the pasta cooking water, a tablespoon at a time and toss until the pasta is evenly moistened. Transfer to a serving platter or bowl and serve warm, room temperature, or cold right out of the refrigerator.