Slice chicken breasts in half through the thickness to create 4 portions total. Cover with plastic wrap and pound with a meat mallet to even out their thickness.
Heat oil in a 12-inch skillet over medium-high heat.
Place flour in a shallow dish. Season both sides of chicken breast portions with salt and pepper then dredge each side in flour, gently shake off excess flour.
Place chicken in skillet (spacing evenly apart) and cook until browned and cooked through, about 4 to 5 minutes per side (chicken should register 165 degrees in center of thickest portion). Note that if skillet seems dry when you turn chicken to second side then drizzle in an additional 1/2 Tbsp olive oil.
Transfer chicken to a plate and cover with aluminum foil to keep warm.
Return skillet to medium heat. Add 2 Tbsp butter and stir to melt.
Add onions and saute until tender and golden brown, about 5 minutes. Add garlic and saute 30 seconds longer.
Slowly pour in 1 1/2 cups chicken broth and the white wine while scraping up browned bits from the bottom of the pan. Stir in thyme leaves.
Bring to a simmer, then reduce heat to medium-low and let simmer until alcohol flavor has cooked off and sauce has reduced to about 1/3 it’s original volume, about 12 – 15 minutes.
In a small bowl whisk together remaining 1 Tbsp chicken broth with cornstarch until smooth. Once sauce in skillet has reduced stir in the cornstarch mixture and cook until thickened, while whisking, about 1 minute longer.
Melt remaining 2 Tbsp butter in sauce. Season sauce with salt if needed and pepper to taste.
Return chicken to sauce in skillet, spoon sauce over and sprinkle with parsley and serve.