Preheat your grill or grill pan over medium-high heat. Season the chicken breasts with olive oil, salt, and pepper on both sides.
Grill the chicken for 5-7 minutes on each side, or until the internal temperature reaches 165°F (75°C) and the chicken is cooked through. Remove from the grill, allow to rest for a few minutes, and then slice into strips.
Bring a large pot of salted water to a boil. Add the Asiago tortellini and cook according to package instructions, usually about 5-7 minutes for fresh tortellini. Once done, drain the tortellini and set aside.
In a saucepan, heat a tablespoon of olive oil over medium heat. Add the minced garlic and sauté for about 30 seconds until fragrant, being careful not to burn it.
Pour in the heavy cream and bring it to a gentle simmer. Stir in the grated Asiago and Parmesan cheese, whisking continuously until the cheese is melted and the sauce is smooth. Season with salt and pepper to taste.
Add the cooked tortellini directly into the sauce, tossing gently to coat the pasta evenly in the creamy Alfredo sauce.
To serve, spoon the tortellini Alfredo onto plates and top with sliced grilled chicken. Garnish with chopped fresh parsley.