Bump the temperature up to medium high and cook, stirring, for about 5 minutes, until the vegetables are slightly caramelized and softened. Add the garlic and cook for about 30 seconds more.
Add the remaining butter to the braiser and let it melt. Add the flour and mix really well, ensuring no clumps remain. Cook, stirring until the flour turns brown and acquires nutty aroma.
Pour in the broth and stir well. Turn off the heat. Taste and adjust the seasonings. Stir in the red wine vinegar.
Add the chicken back, nestling the pieces inside the vegetable mix.
Cover the braiser and transfer to an oven preheated to 350 F. Braise for 2 1/2 to 3 hours.
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