Fill a large pot 3/4 full with water. Add 1/4 cup of the salt and bring to a boil.

Rub the cut sides and bottoms of the artichokes with the lemon wedges and carefully add them to the boiling water. Weight the artichokes as necessary with a heavy dish or bowl, and simmer, partially covered, until the bottoms are just tender and can be pierced with a sharp knife and an outer leaf pulls out easily, about 15 minutes. Drain the artichokes upside down in a colander until cool enough to handle.

Cut the artichokes into quarters and discard the prickly purple leaves and hairy choke.

In a bowl, combine the olive oil, vinegar, garlic, parsley, salt, and pepper. Add the artichoke quarters and toss to coat. Let marinate for 2 to 4 hours, turning occasionally.

Preheat the grill to medium-high.

Remove the artichokes from the marinade and grill, turning, until warmed through and lightly charred around the edges, about 5 minutes. Place on a platter and top with the crumbled cheese. Drizzle with the grilled lemon mayonnaise and serve additional mayonnaise on the side for dipping.

Preheat the grill to medium-high.

Place the lemon halves, cut sides down, on the grill and cook until charred, about 1 minute.

Squeeze the lemon juice into the bowl of a blender or food processor. Add the egg and 1 tablespoon water and process for 20 seconds. With the processor running, slowly pour in the olive and vegetable oils through the feed tube and process until emulsified, adding additional water 1 tablespoon at a time to thin the mayonnaise to drizzling consistency. Add the salt and cayenne pulse to blend.

Adjust the seasoning to taste, transfer to an airtight container, and refrigerate for at least 30 minutes before using. (Mayonnaise will keep refrigerated for up to 24 hours.)