Serve in wedges as an appetizer with drinks, or serve whole, with soup or salad, as a light lunch.
Cut away leaves of artichokes, leaving the hearts. Scoop out and discard chokes. Cut each heart in half lengthwise. Place in a steamer basket set in a large saucepan filled with 2 to 3 inches of water. Bring to a boil; lower heat, cover, and steam until hearts are barely tender, about 15 minutes. Refresh in ice water, drain well, and dry thoroughly. Cut each half into 8 wedges.
In a medium bowl, whisk together 2 tablespoons of the olive oil, the lemon juice, and salt and pepper to taste. Carefully stir in artichoke hearts, making sure they are thoroughly coated. Let marinate until ready to use.
In a large skillet, heat remaining 1/4 cup olive oil over medium heat. Add shallots and cook, covered, just until tender, 6 to 7 minutes. Do not allow to brown. Spread on a platter to cool.
Roll out puff pastry to 1/8 inch thick, and cut out four circles five inches in diameter. Fold in 1/2-inch edge, pleating as you go along, and place on parchment-lined baking sheets. Refrigerate for 30 minutes, or until pastry is cold.
Heat oven to 375 degrees. Remove shells from refrigerator. To assemble tarts, spread shallots on shells, making a layer about 1/4 inch thick and leaving a border of about 1/4 inch. Arrange artichoke wedges on top of shallots. Brush edges of dough with egg wash. Bake until crust is golden brown and shallots are lightly colored, about 15 minutes. Serve immediately, drizzled with Mark Peel's Pesto.