Slice chicken breasts in half horizontally to make 4 thin cutlets. Pat dry. Season both sides with salt, pepper, garlic powder, and smoked paprika.
Bring a large pot of salted water to a boil.
Heat olive oil and butter in a large skillet over medium-high until butter foams.
Add chicken cutlets in a single layer. Do not move. Cook 4–5 minutes until golden underneath.
Flip once. Cook 3–4 minutes until internal temperature reaches 74°C (165°F).
Transfer to a cutting board. Rest 5 minutes. Do not cut yet.
Cook pasta in boiling salted water until just under al dente. Reserve ½ cup pasta water before draining.
In the same skillet, reduce to medium heat. Add olive oil and garlic. Cook 1 minute, scraping up browned bits.
Add cherry tomatoes. Cook 3–4 minutes, pressing gently until they burst.
Add drained pasta, pesto, and ¼ cup pasta water. Toss over medium heat 1–2 minutes until glossy and evenly coated.
Remove from heat. Add lemon juice and Parmesan. Toss once more.
Slice rested chicken and lay over the pasta.
Finish with Parmesan shavings, fresh basil, lemon wedge, cracked pepper, and a drizzle of olive oil. Serve immediately.