Season chicken with salt and pepper. Add 1 cup flour to a medium shallow bowl. Dip chicken in flour, shaking off any excess.

In a large cast iron skillet, or heavy bottom skillet over medium high heat, heat oil. Working in batches, if necessary, add chicken and cook until light golden-brown, about 1-2 minutes per side. Transfer to a plate and set aside. Repeat with remaining chicken.

Reduce heat to medium. Add onions and cook, stirring every few minutes, and adding a splash of broth, if necessary to prevent it from coloring too much in any one place (reduce the heat if onions begin to darken too quickly). Cook, scraping up any brown bits from the bottom of the pan, until onions are dark golden-brown, about 25-30 minutes.

Stir the 2 tablespoons of flour into the onions and cook until no lumps remain, about 1 minute. Slowly stir in broth, vinegar and 2 teaspoons of fresh thyme, scraping up brown bits from bottom of pan, as needed, until combined.

Return chicken to skillet. Cover and simmer until sauce thickens and chicken reaches an internal temperature of 165°(F) on an instant-read thermometer, about 3-5 minutes. Top each chicken breast evenly with Gruyere. Cover and cook until cheese is melted, about 1 minute.