Season the chicken breasts with salt and pepper. Dredge in flour, shake off excess and set aside.
Melt 4 tablespoons of butter and 5 tablespoons of olive oil in a large skillet or a saute pan over medium-high heat. When butter and oil start to sizzle, add the chicken breast and saute for 2-3 minutes per side, until the chicken is nicely browned and juices run clear. Take the pan off the heat, remove the chicken and set aside.
Turn the heat down to medium. Return the pan to heat. Add the remaining 2 tablespoons of butter. Once the butter starts to sizzle, add two tablespoons of flour left over from dredging chicken and keep stirring until the flour is well incorporated and looks golden brown.
Pour the chicken stock and lemon juice into the pan. Return the pan to the heat and bring to boil, scraping up brown bits from the pan for extra flavor. Add the pesto and stir. Taste and adjust the seasoning, if needed.
Add the sliced cherry tomatoes and chopped basil, and continue sauteing over medium heat for 1-2 minutes, until the tomatoes soften up a little.
Put the chicken breasts back into the pan, spoon the sauce over the chicken breasts, and simmer over low heat for about 2 minutes to bring the chicken back to temperature.
Remove from heat and serve immediately with pasta, rice or steamed vegetables. Garnish with grated Parmesan cheese and basil leaves.