To clean each artichoke, pull off the tough green outer leaves until you get to the tender leaves that are mostly yellow. Peel stem and slice about 1" off top of artichoke. Spread open leaves and scoop out fibrous white choke, using a spoon. Rub artichokes all over with lemon to prevent discoloration.
Stir together mint, garlic, and salt in a mixing bowl. Pack approximately 2 tbsp. mint mixture between leaves and rub a little in the center of each artichoke.
Pour 1⁄2 cup of the oil into a medium saucepan. Place artichokes stem side up in saucepan and pour wine and remaining oil on top. Cover and cook over medium-low heat until tender (check for tenderness at thickest part of the stem), about 40 minutes. Season to taste with salt and pepper.
Transfer artichokes to a platter and allow to cool completely. (They are best when served at room temperature.) Continue to cook juices until reduced by half, then serve as a sauce, drizzled over artichokes. Garnish with red leaf lettuce and watercress. Store artichokes with reduced juices in refrigerator for up to 1 week.