How to Make It
Combine 2 cups water and 2 tablespoons juice in a large bowl. Cut stem of each artichoke to within 1 inch of base; peel stems. Remove bottom leaves and tough outer leaves, leaving tender heart and bottom. Cut each artichoke in half lengthwise. Remove fuzzy thistle from bottom with a spoon. Thinly slice each artichoke heart into 1/8-inch-thick slices; place in lemon water.
Combine remaining 1 1/2 tablespoons juice, thyme, pepper, salt, and anchovy in a small bowl. Gradually add oil, stirring constantly with a whisk until combined.
Drain artichoke hearts; pat dry with paper towels. Combine sliced artichoke hearts, anchovy mixture, parsley, and mushrooms in a large bowl; toss gently. Divide artichoke mixture among 6 small plates; sprinkle each salad with about 2 teaspoons cheese.