Cut off and discard artichoke stem and all but 1 1/2 in. from leafy top. Pull off leaves down to pale yellow ones. Cut off all green parts from outside. Halve artichoke through stem and use a pointed teaspoon to dig out fluffy, fibrous choke. Cut artichoke into 1/4-in. dice.
Cook fresh (or thawed frozen) artichoke, shallot, 1 tbsp. oil, and 1/4 tsp. salt in a small saucepan over medium-low heat, covered, 5 minutes, stirring occasionally. Add minced garlic; cook until artichoke is tender, 5 minutes more. Let cool.
To artichoke mixture add 2 tbsp. oil, the lemon juice, 1/4 tsp. pepper, the chopped parsley, and olives. Stir; let relish sit at least 1 hour.
Preheat broiler with rack set close to heat. Broil baguette slices on a baking sheet, turning once, until light golden, 3 to 4 minutes total. On 1 side, lightly rub toasts with garlic clove, brush with remaining 1 tbsp. oil, and sprinkle with salt and pepper.
Spread toasts with cheese, leaving a corner uncovered, and top with relish. Set a parsley leaf on each.
Make ahead: Topping, up to 1 day, chilled.