Pull and snap off darker outer leaves of artichokes until you reach the pale green-yellow tender inner leaves. Slice off top third of the artichoke. Trim very end of stem and then peel outer layers of stem* with a paring knife or vegetable peeler. (The outer layer of the stem is super fibrous, but the inner, lighter heart is sweet and succulent.)
Slice whole artichoke in half lengthwise (don't use a carbon-steel knife, or artichoke will discolor) and rub the whole exterior with one of the lemon halves. Scoop out hairy choke with a spoon, or slice it away with a paring knife. Squeeze some lemon juice into choke space.
Place an artichoke half cut side down on work surface and slice it lengthwise as thinly as you can. If you have a mandoline slicer, this is the perfect time to use it. Repeat with other artichoke halves.
Put sliced artichoke in a bowl. Squeeze in juice of remaining 3 lemon halves (try to retrieve and discard seeds!) and add 1/2 tsp. salt, lots of twists of black pepper, the chile flakes, mint, parsley, chives, chive blossoms (if using), almonds, and Parmigiano and toss. Taste and adjust the seasoning so salad is lively and well balanced, then drizzle with olive oil. Toss salad again, taste, and serve.
*Trim the base of the artichoke as well.