Place chicken breasts between 2 pieces of waxed paper and pound to an even 1/2-inch thickness with a meat mallet (or use a rolling pin). Trim any excess fat or ragged edges. Season both sides with salt and pepper.
Melt butter with oil in a large skillet over medium-high heat. When foam subsides, cook chicken until browned and cooked through but still slightly springy to the touch, 3 to 4 minutes on each side. Transfer to a plate and cover loosely with foil.
Pour off all but 1 Tbsp. fat from skillet. Add broth, increase heat to high and boil, stirring to pick up any browned bits on bottom of skillet, until slightly reduced, about 2 minutes. Add zucchini and squash; sprinkle with salt and pepper. Cook, stirring often, until softened, about 3 minutes. Stir in tomatoes and thyme. Cook just until tomatoes are warmed through, about 2 minutes. Season with additional salt and pepper, if desired.
Using a sharp knife, cut chicken into thick slices on a diagonal. Serve warm with a spoonful of vegetables on the side.