Squeeze the lime juice (from the now cooled lime) through the feed opening of a running blender. With the blender still running, add the vinegar, corn syrup, shallot and rosemary, 1 at a time, and pour the oil in a slow steady stream into the vinegar mixture very slowly to emulsify. Remove to a container and season with salt and pepper, to taste.
For the spring greens:
Tear the greens into bite-sized pieces, and remove the stems if using arugula. (Make sure greens are no longer wet from being rinsed.) Toss greens with Parmesan, season with salt and pepper, to taste, and set aside.
For the filling:
Heat the clarified butter over medium heat in a large saute pan. Add the onion and cook for about 15 minutes until translucent and beginning to caramelize, then add the portobellos and white mushrooms to the onions. Cook for another 10 minutes then reduce heat to low and add the tomatoes, chives, salt, and pepper, and allow flavors to integrate.
For the vol-au-vent (tart pastry):
Preheat oven to 375 degrees F.
Roll out puff pastry dough, and using a 4-inch circle cutter (aka biscuit cutter), cut out 8 dough rounds. Place 4 of the dough rounds on a lightly greased baking sheet. Using a 3-inch circle or biscuit cutter, cut a concentric circle in the middle of the remaining 4-inch dough rounds to create a 1/2-inch wide circle of dough. These 1/2-inch wide circles of dough are to be used without the 3-inch pieces from the center. Place 1 of these 1/2-inch wide circles of dough on top of each of the rounds already on the baking sheet, matching up the edges. Brush egg wash over the surface of each of the paired dough circles and rounds. Then use a fork to prick through the dough around the perimeter (that is, through each dough circle and into the accompanying round beneath it). Place in the oven and bake until golden brown, about 6 to 8 minutes. The center of each will rise. Remove pastry shells from oven and allow to cool slightly.
Using the tip of a butter knife, break through the risen center part of each pastry shell to make a crater. Spoon some of the mushroom mixture into each of these craters. Place some greens on top of each tart and drizzle with lime balsamic vinaigrette.