Heat the butter and olive oil in a cast iron skillet over medium heat, then add the thinly sliced onions with the sugar and a pinch of salt and pepper. Cook, stirring, for 40 minutes until deeply golden and jammy with dark caramelized bits.

Stir in the Worcestershire sauce and cook for 1 additional minute to coat the onions. Scrape up any browned bits for extra flavor.

Cool the caramelized onions completely before mixing. This helps the dairy filling stay thick instead of melting out.

In a bowl, mix the cooled onions with the shredded gruyere, softened cream cheese, and fresh thyme until evenly combined. Season with additional salt and pepper if needed for balance.

Cut a pocket into each boneless skinless chicken breast and season the outside with salt and pepper. Keep the pocket intact so the filling stays inside.

Stuff each chicken pocket generously with the onion-cheese mixture, then secure the opening with toothpicks. Press gently so the seam closes as much as possible.

Sear the stuffed chicken in the cast iron skillet for 3 minutes per side until browned. Keep the heat medium-high for color without burning.

Transfer to a sheet pan and bake at 375F for 20-25 minutes until the chicken is cooked through. If needed, bake longer in 3-minute increments to reach doneness.

Rest the chicken briefly before serving. This lets the molten cheese set slightly so it stretches neatly when sliced.