Cut each chicken breast in half width-wise so you have 4 thin chicken cutlets. Pound chicken cutlets flat to about 1/2-in thickness between two sheets of wax paper using a meat mallet or a rolling pin.
Place flour in a shallow bowl and season with salt and pepper to taste. Dredge chicken cutlets in flour until evenly coated, shaking off excess
Place 1 tablespoon butter and 1 tablespoon olive oil in a large skillet and heat over medium-high heat. Add chicken in a single layer to pan (work in batches if needed) and sear 3-4 minutes on first side. Flip chicken and add remaining 1 tablespoon butter and cook an additional 3-4 minutes until chicken registers 165F. Transfer to a cutting board and keep warm.
Add remaining olive oil to same pan and heat over medium heat. Add shallot and cook 2-3 minutes until tender. Add garlic and cook 1 minute until fragrant.
Deglaze pan with white wine, using a wooden spoon to scrape up any browned bits. Add chicken stock and bring to a simmer. Reduce heat to medium-low and allow to simmer 3-5 minutes until slightly reduced.
Remove sauce from heat and stir in lemon zest, juice, and capers. Season sauce with salt and pepper to taste and pour into a small bowl.
Cook tortellini in salted boiling water until al dente. Drain and transfer tortellini to same skillet you made the sauce in.
Slice chicken into thin strips and arrange over tortellini. Drizzle warm lemon caper sauce over top and garnish with parsley. Serve immediately while warm and enjoy!