Heat a non-stick pan over medium heat.
Season the chicken on both sides with salt and pepper.
When the pan is hot, add half of the oil. Sear the chicken to a golden brown on both sides, then remove from heat and set aside on a plate.
Add the butter and remaining oil to the pan, then turn the heat down to medium-low. Add the onions and cover with a lid. Cook, stirring occasionally until the onions turn golden brown and soft (about 20 minutes).
Once golden brown, add the garlic and sauté for 1 minute, then stir in the flour and cook for 1 minute longer. Stir in the broth, thyme, and add the chicken into the pan. Cover with a lid.
Bring to a low simmer for about 10 minutes. Taste and adjust the seasoning with salt and pepper. The sauce consistency should coat the back of a spoon. If it's too thick, you can thin it out with a little more broth; if it's too thin, you can mix a few teaspoons of sauce in a bowl with a little flour to form a paste, and add it back into the sauce a little at a time until it is thickened to the proper consistency.
Top the chicken with cheese and cover with a lid until melted. Remove the cooked thyme sprigs before serving.
Serve with crusty bread. Garnish with fresh thyme sprigs.