Heat a nonstick pan on 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 golden brown on both sides then remove from heat and set aside on a plate.
Add the butter and remaining oil to the pan and turn the heat down to medium-low. Add the onions and cover with a lid. Cook and stir 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.