Pat the chicken dry with a paper towel, then slice it in half horizontally (if not already butterflied). Sprinkle the chicken on both sides with the salt, garlic powder, and onion powder.
Add the butter and olive oil to a large skillet and heat over medium heat. Add the chicken and cook until browned on both sides, about 4 minutes per side. When the internal temperature is 165°F or the center is no longer pink, remove to a plate and rest for 3 minutes.
Meanwhile, in a medium bowl, stir together the basil pesto and cream with a pinch of kosher salt. When the chicken has rested, gently toss the chicken breasts in the sauce and serve immediately with a drizzle of the sauce over the chicken.