Adjust rack to middle position and preheat oven to 375°F.
Place the garlic in the center of a square of tinfoil and drizzle a third of the olive oil over the cloves. Scrunch it up into a foil pouch and place into the oven, roasting for 35-40 minutes until tender. Let cool completely in the foil.
Place roasted garlic cloves into the bowl of a food processor with the mayonnaise, juice from half a lemon (if doubling or tripling this recipe, adjust the amount), and the remaining olive oil. Puree until the mixture is as smooth as possible. It may still have a few chunks, that’s okay. Transfer to a small bowl and mix in capers and half the parsley. Season with freshly ground black pepper, to taste.
To clean the artichokes, start by filling a large bowl with water. Squeeze the remaining lemon juice into the water, placing the entire lemon half into the bowl as well. Peel off any of the harder outer leaves, starting at the base of the choke (I usually go around the artichoke about 3 times until I have peeled off all harder outer leaves). Next, trim the root of the stem, if any. With a paring knife or peeler, peel the stem of the artichoke, exposing the pale green center of the stem. Cut the artichoke in half lengthwise and then trim off half an inch off the top of the artichoke. Place immediately into the lemon water to prevent browning.
Once all the artichokes have been prepped, drain and pat dry with paper towels, drying them well. Place into a heavy bottomed sauce pot (make sure not to overcrowd - If you need to get a wider pot, please do so) and pour frying oil over the artichokes, covering by 1-2 inches or halfway up the side of the wide pot or pan.
Turn heat to medium and bring to oil to a simmer. Reduce the heat to low, cover the pot, and cook for 12 minutes. Then uncover the pot, raise the heat to medium-high, and cook for another 3-5 minutes, until artichokes are crispy.
Remove artichokes from the oil with a slotted spoon and drain on paper towels. Sprinkle immediately with Kosher salt. Place onto a serving dish and sprinkle with remaining parsley. Serve with roasted garlic caper aioli, for dipping. TIP: If you have extra lemon, squeeze a wedge of juice on top of the artichokes.