Heat a Dutch oven with evoo, over medium heat. Brown eggplant, onions, leeks, mushrooms, and carrots, in batches, adding more evoo as needed and seasoning with salt and pepper, until vegetables are golden and caramelized.
Preheat oven to 350°F.
Add vegetables and garlic back to the pan with artichokes, olives, saffron, lemon slices, thyme, parsley stems, and wine. Cover and braise in oven at 350°F for 1 hour until vegetables are tender. Strain out garlic and discard (all the flavor is left behind).
Serve barigoule with braising liquid and garnish with fresh parsley and additional lemon slices.