Place chicken breasts between two pieces of plastic wrap. With a meat mallet or heavy skillet, pound chicken to an even 3/4-inch thickness. Season with salt and pepper.
Heat the oil in a large nonstick skillet; cook the chicken until golden brown and cooked through, 3 to 4 minutes per side. Slice each piece across the grain into 4 or 5 strips; cover to keep warm.
Meanwhile, cook pasta according to package instructions. Reserve 1/4 cup cooking water. Drain pasta and return to pot.
In a small saucepan, heat pesto over low until liquefied. Stir in cream; heat until warm. Toss pasta with reserved cooking water and half the sauce. Divide among shallow bowls; top with chicken. Drizzle with remaining sauce.