Prep first. If you begin mincing the shallots and garlic when it's time to toss them in the skillet, you could overheat the oil and scorch the food.
Slice through each chicken breast horizontally to form 2 thin cutlets. Warm a 12-inch skillet over medium-high heat; season chicken on both sides with salt and pepper. Add 1 Tbsp. oil to skillet; swirl to coat. Add chicken and cook, undisturbed, until golden, about 2 minutes. Turn and cook until just cooked through, 1 to 2 minutes longer. Transfer to a platter and tent with foil to keep warm.
Add remaining 2 tsp. oil and shallot to skillet; sauté for 1 minute. Add wine and cook, stirring to release brown bits from bottom of skillet, until liquid has reduced by about half, 1 to 2 minutes. Add thyme and remove skillet from heat. Add butter and swirl until sauce is creamy. Spoon sauce over chicken and serve.