Bring a large pot of water to a boil.
Meanwhile, place the chicken breasts on a cutting board and cover with parchment paper or plastic wrap. Using a meat mallet or the bottom of a heavy skillet, pound the chicken until it is a uniform 1/2-inch thickness. Pat dry with paper towels. Season both sides with 1 teaspoon salt and 1/2 teaspoon of black pepper. Set aside.
When the water reaches a boil, add the uncooked pasta and a big pinch of salt (about 1 tablespoon) and cook until al dente, according to package directions. When the pasta is tender, reserve 1/2 cup of the pasta water. Drain the pasta. Set aside.
Heat the oil in a large, deep skillet set over medium-high heat. Add the chicken in a single layer and cook until golden brown and cooked through, about 4 minutes per side. Transfer to a clean plate and set aside.
When the pasta is tender, reserve ½ cup of the pasta water and drain the pasta. Set aside.
Reduce the heat in the skillet to medium low. Melt the butter, then add the heavy cream and pesto and stir, scraping up any brown bits on the bottom of the pan. Add the Parmesan and stir until melted and well combined, about 2 minutes.
Add the pasta, reserved pasta water, 1 teaspoon salt, and frozen peas. Cook, stirring, until the pasta is well coated in the sauce and the peas are tender, about 3 minutes. Taste and add more salt, as desired.
Remove from the heat and stir in the basil. Slice the chicken breasts and nestle on top of the pasta, then garnish with more grated Parmesan and fresh basil. Enjoy!