Preheat the oven to 325 degrees . Add the flour to a shallow bowl. In a dutch oven, heat 2 tbsp. EVOO over medium-high. Season the drumsticks, then coat in the flour, shaking off the excess. Cook until browned on all sides, 10 minutes. Transfer to a plate; discard the cooking fat.
Add the yellow onion, carrots and remaining EVOO to the pot. Lower the heat to medium and cook, stirring, until lightly browned, 5 minutes. Add the wine, drumsticks and thyme. Press a parchment round on top, cover and bring to a simmer. Transfer to the oven; cook, turning the meat occasionally, until tender, 1 1/2 hours.
Meanwhile, in a large skillet, cook the bacon over medium heat, stirring, until browned, about 5 minutes. Using a slotted spoon, transfer to a paper-towel-lined plate. Transfer 1 tbsp. of the bacon fat to a bowl and leave the remainder in the skillet. Add the mushrooms, cover and cook over medium heat until tender, about 5 minutes; transfer to a bowl. Add the reserved 1 tbsp. bacon fat and the cipollini onions to the skillet, cover and cook over medium-low heat, stirring occasionally, until tender, about 20 minutes; add the the mushrooms.
Remove the drumsticks from the oven and transfer to a cutting board. Strain the liquid into the skillet; discard the solids. Bring to a boil and cook until thickened, 10 minutes; season. Lower the heat to low. Shred the drumsticks, discarding the skin. Transfer to plates with the sauce, bacon and vegetables.