Place chicken breasts between two sheets of parchment and lightly pound to an even thickness so they cook evenly.
Season both sides of the chicken with salt, pepper, and garlic powder.
Heat 2 teaspoons olive oil and 2 teaspoons butter in a large skillet over medium-high heat.
Cook the chicken in the hot skillet until golden and cooked through, about 5–6 minutes per side depending on thickness. Transfer cooked chicken to a plate and tent with foil to keep warm.
Reduce heat to medium and add the remaining butter and oil to the same pan.
Add the minced garlic and cook, stirring, until fragrant, about 1 minute.
Add the halved grape tomatoes and cook for about 2 minutes, until they begin to soften.
Turn off the heat and stir in the shredded basil. Taste and add extra salt or pepper if desired.
Return the chicken to the pan and spoon the tomato–garlic mixture and pan juices over the chicken before serving.