Start by filleting your chicken breasts to create thinner, more evenly-sized pieces. Place each breast on your cutting board and use a sharp knife to cut horizontally through the middle, creating two thinner cutlets from each breast. This technique, called butterflying, ensures the chicken cooks quickly and evenly without drying out. Season both sides of each piece with salt and pepper, pressing gently so the seasoning adheres well.
Heat your large skillet over medium heat and add one tablespoon of olive oil, swirling to coat the surface. Once the oil shimmers and moves freely around the pan, carefully add the seasoned chicken pieces. Cook without moving them for about 4-5 minutes until they develop a beautiful golden-brown crust, then flip and cook the other side for another 3-4 minutes until cooked through. Remove the chicken to a clean plate and set aside - don't worry if it looks a bit pale, it'll finish cooking in the sauce later.
Reduce your heat to medium-low and add the remaining tablespoon of olive oil to the same skillet. Add the halved grape tomatoes and minced garlic, stirring gently to coat with oil. Let them cook for about 5 minutes, stirring occasionally, until the tomatoes start to soften and release their juices. You'll notice they begin to lose their firm shape and become slightly caramelized - this concentrates their sweetness beautifully.
Pour in the heavy cream and add the pesto to the skillet with the tomatoes and garlic. Stir everything together gently, scraping up any browned bits from the bottom of the pan - those add incredible flavor. Allow the sauce to come to a gentle simmer, then taste and adjust with additional salt or pepper if needed. The sauce should be creamy and well-combined, with the pesto evenly distributed throughout.
Return the seared chicken to the skillet, nestling it into the creamy sauce. Spoon sauce over the top of each piece and let everything simmer together for 2-3 minutes until the chicken is heated through and the flavors meld. The internal temperature of the chicken should reach 165°F for food safety. Serve immediately while hot, spooning extra sauce over each portion.