In the bowl of a food processor fitted with a metal blade, add the basil, Parmesan, pine nuts, half of the garlic, salt and black pepper and pulse to combine. With the food processor running, drizzle in the 1/3 cup olive oil until the sauce comes together and is smooth. Set aside.
Slice the chicken breast in half horizontally so you have 8 thinner cutlets. Sprinkle both sides of each cutlet with salt and pepper.
Heat 1 tablespoon each butter and olive oil in a large cast iron skillet over medium-high heat. Add half of the cutlets and cook through, 3 to 4 minutes per side. Remove to a plate. Repeat with the remaining cutlets.
Reduce the heat to medium and add the remaining 3 tablespoons butter and 1 tablespoon olive oil along with the red onion and remaining sliced garlic. Cook, stirring frequently, 1 to 2 minutes. Add the rainbow chard and cook, stirring, until lightly wilted, about 1 minute. Add the lemon juice and stir for an additional 30 seconds. Remove to a plate.
Without cleaning the skillet, add the cream and heat for 1 minute. Stir in about a third of a cup of the pesto and return the chicken to the pan. Bring to a simmer and allow to thicken, 1 to 2 minutes. (Refrigerate the remaining pesto in an air tight container for up to 5 days.)
Make a bed of rainbow chard on a large platter. Top with the chicken cutlets and drizzle the sauce over the top. Sprinkle with the tomatoes and serve.