Preheat oven to 350°F. Season riblets with salt and pepper. Heat oil in Dutch oven over medium heat. Brown riblets on all sides. Transfer to a plate. (6 - 8 minutes)
Deglaze pan with a small amount of wine. Saute mirepoix (onions, carrots, celery) for 4 - 6 minutes, stirring occasionally, until softened. Add the garlic and cook until just fragrant (1 - 2 minutes).
Add remaining wine, bring just to a boil and then reduce heat to simmer until wine is reduced by half (15 - 20 minutes).
Return the riblets, add the herbs, stir in broth (or stock), cover Dutch oven and place in oven. Cook for 90 - 120 minutes, or until meat is fall-off-the-bone tender.
Remove from oven, let rest for 5 - 10 minutes, and serve.