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.
