Preheat oven to 400°F. Then lightly grease a 9x12" baking-dish. Put aside.
Melt butter in a large-skillet over medium-high-heat. Then add onions and season with pepper, salt and half of fresh chopped thyme and cook for 5-8 minutes till tender.
Reduce heat to moderate and cook, while stirring occasionally till onions are caramelized and jammy for 15 minutes.
Then pour ¼ cup of broth in the pan becomes too dry and keep stirring till onions are browned.
OPTIONAL: Pour in balsamic vinegar, white wine or sherry to deglaze the pan and add some extra tastes to the onions. Let cook for a moment before the sauce thickens slightly.
Stir in garlic and cook till aromatic for a few minut. Then turn heat off and let cool slightly.
Cut chicken breasts in half horizontally to make pockets. Season chicken with ground thyme, garlic powder, pepper and salt.
Spoon 1-2 tbsp of caramelized onions to every pocket. Stuff each breast with all the cheeses and seal chicken together with toothpicks.
Transfer the remaining caramelized onions in the skillet into the prepared baking-dish. Mix in remaining ¼ cup of broth. Put aside.
Heat oil in exactly the exact same skillet over medium-high-heat. Sear the stuffed chicken breasts for 4 minutes each side till browned. (They won't be completely cooked. You may finish off them in the oven).
Top with remaining chopped thyme and then bake for 15-20 minutes till completely cooked.
Discard toothpicks. Serve with pan juices and onions.