In a nonstick skillet, melt 1 tbs. butter into 2 tbs. EVOO. Add the anchovy filets and garlic; cook until the anchovies melt. Add the bread and cook over medium heat until toasted on all sides. Remove from heat and set aside.
Prepare each artichoke by cutting 1-inch off the top, and snipping the pointed tips off the remaining leaves, with scissors. Cut the stem off at the base of each artichoke so they can sit level in the pot. Trim the bottom of the stems about ½ inch.
In a Cuisinart or chopper, add the toasted bread and garlic, locatelli cheese, 2 tbs. EVOO, parsley, 1 tbs. salt, cracked black pepper, and red pepper flakes. Pulse until well chopped/combined.
Grasp the base of the artichoke and with a firm hand, hit the top of the artichoke, cut-side-down, on to your counter or cutting board. Spread apart the leaves to make room for the stuffing. Use your hand to stuff the breadcrumb mixture into the artichoke, making sure to get in between the leaves.
Heat 3 tbs. EVOO in a large stock pot. Place the artichokes in the pot, arranging the cut-stems in between, and cook for 1 minute. Add the remaining 2 tbs. of salt, and the water. Bring to a boil, cover and cook for 15 minutes. Reduce heat and simmer covered for at least 2-hours (can cook all day!).
Serve individually with one stem, and some of the pot liquor. Be sure to provide a plate for discarding the leaves.