1. Heat oven to 375 degree F. Line 2 large cookie sheets with parchment paper. Set aside.
2. To make tarts, toss asparagus with 1 tablespoon olive oil, herbes de Provence, salt, and pepper in a large bowl. Set aside.
3. Heat remaining tablespoon olive oil in a large skillet over medium-low heat. Add onions and cook, stirring frequently, until browned and very tender, about 20 minutes. Set aside.
4. On a lightly floured surface, roll out one sheet of puff pastry (keep remaining sheet refrigerated) to form a 10x15-inch rectangle (trim edges if needed to make them straight). Trim a 3-inch wide piece from both the bottom and right side of pastry so that you are left with a 7x12-inch rectangle. Cut each trimmed 3-inch piece lengthwise into six 1/2-inch strips, then cut 7x12-inch rectangle into three 4x7-inch pieces. Brush a 1/2-inch border around pastry with egg, then place one 1/2-inch strip on each of the four sides to form a crust. Repeat. Continue process with second sheet of pastry to end up with 6 tarts. Divide onions down the center of each pastry. Neatly arrange whole pieces of asparagus, with flower ends facing the same direction, over onions. Bake until asparagus is tender and pastry is puffed and golden brown, 25 to 30 minutes. Remove from oven and immediately sprinkle shaved Parmesan cheese over top.
5. To make chicken paillard, place chicken breasts, one at a time, in a large resealable plastic bag and pound until thin. Transfer to a large bowl. In a small bowl combine lemon juice, onion, mustard, sugar, salt, and pepper. Whisk in olive oil until creamy. Whisk in tarragon. Separate and set aside 1/4 cup of dressing. Combine remaining dressing with chicken and refrigerate 20 minutes.
6. Oil a grill pan over medium-high heat. In batches, grill chicken until cooked through, about 2 minutes per side. Transfer chicken to a serving platter and drizzle with remaining dressing. Serve with tarts. Makes 6 servings.