In a large skillet, heat 1 tablespoon of olive oil over medium heat.

Add the sliced onions and cook, stirring frequently, for about 15–20 minutes.

Stir in the minced garlic, thyme, balsamic vinegar, salt, and pepper, then cook for an additional 2–3 minutes.

Remove from heat and let the mixture cool slightly.

While the mixture cools, preheat your oven to 375°F (190°C).

Slice a pocket into each chicken breast, being cautious not to cut all the way through.

In a bowl, combine the cooled caramelized onion mixture with the shredded Swiss cheese and softened cream cheese until well mixed.

Spoon the filling into each chicken breast pocket, securing them with toothpicks if necessary.

In the same skillet, heat the remaining tablespoon of olive oil over medium-high heat.

Add the stuffed chicken breasts and sear for 3–4 minutes on each side.

Carefully transfer the seared chicken to a baking dish and sprinkle breadcrumbs on top if desired.

Bake for 20–25 minutes until the chicken is cooked through and reaches an internal temperature of 165°F (74°C).

Remove toothpicks if used and garnish with fresh parsley before serving.