In a large nonstick skillet, heat the oil over medium heat. Add the onions and 1/4 teaspoon salt. Increase the heat to medium-high and cook until the onions start to brown.
Once browned, turn the heat to medium-low. Add the balsamic vinegar, Worcestershire sauce, and thyme. Cook for 20 to 25 minutes, stirring occasionally, until the onions are very tender.
Season the chicken breasts with salt and pepper. In a shallow bowl, whisk together the flour, salt, and paprika.
In another skillet, melt 1 teaspoon of butter over medium heat. Dredge the chicken in the flour mixture and cook for 3 minutes on each side until lightly browned. Set the chicken aside.
Combine the chicken broth and 1 tablespoon of the leftover flour mixture. Pour this into the skillet, whisking until the sauce thickens.
Return the chicken to the skillet, topping it with the caramelized onions and Gruyère cheese. Cover and cook on low for 3 to 5 minutes until the cheese is melted and bubbling.