Heat 2 tablespoons of EVOO in a Dutch oven over high heat.
Season pork liberally with salt and pepper. Place into pot and brown on all sides. Remove and reserve.
Into the same pot, add 2 more tablespoons of EVOO, if needed. Add quartered onion, carrot, whole garlic cloves and cook until soft. Put the pork back into the pot and fill with enough water to cover meat. Add lemon and herb sachet. Cook over medium heat for 1 1/2-2 hours, until pork is cooked through.
Remove pork and strain braising liquid back into pot. Add anchos and cook over medium heat to soften. Purée liquid in a food processor and reserve. Shred pork using two forks.