Place pine nuts, garlic, anchovies, and crushed red pepper flakes in a large bowl; mix and crush using the back of a spoon. Add breadcrumbs and cheese, squeeze 2 lemon halves over filling mixture, and mix to combine. Stir in parsley and mint; drizzle with enough olive oil to moisten mixture. Set filling aside.

Fill a medium bowl halfway with water. Squeeze remaining 2 lemon halves into water and add lemon halves to water. Set acidulated water aside.

Cut 1 inch from top of an artichoke. Spread leaves and remove choke using a melon baller. Peel off outer leaves and discard. Cut off bottom stem and discard. Transfer artichoke to acidulated water and repeat process with remaining artichokes.

Preheat oven to 400 degrees. Remove artichokes from water and pat dry; generously stuff artichokes with filling and transfer to a medium ovenproof saucepan. Add wine to saucepan and add enough water so that the liquid comes 1 inch up the sides of the artichokes. Cover saucepan and bring to a boil over medium-high heat. Reduce heat to a simmer and cook until artichokes are tender, about 20 minutes.

Uncover saucepan and transfer to oven; cook until stuffing is toasted, about 7 minutes. Remove artichokes from saucepan and drizzle with olive oil. Season with salt and pepper and serve with a spoonful of their cooking liquid, if desired.