For squash blossoms
For tempura batter and frying
Stuff squash blossoms:
Preheat oven to 250°F.
Toss tomatoes with oil and arrange, cut sides up, in 1 layer in a shallow (1-inch-deep) baking pan. Bake in middle of oven until slightly shriveled but not completely dried or browned, about 1 1/4 hours. Transfer pan to a rack to cool.
Stir together goat cheese, cream, pumpkin seeds, basil, salt, and pepper in a bowl and fold in half of oven-dried tomatoes.
Spoon 1 rounded tablespoon of this filling into each blossom and twist ends of petals gently to close. Chill, covered, until ready to fry.
Prepare vinaigrette and shaved squash:
Purée vinegar, shallot, oil, pepper, and salt in a blender until smooth and emulsified.
Slice squash paper-thin (lengthwise) using slicer, then overlap squash slices decoratively on 4 plates to cover surface.
Make tempura batter and fry blossoms:
Heat 2 inches of oil in a 3-quart saucepan to 350°F on thermometer.
Set a bowl in a larger bowl of ice and cold water, then whisk together flour and salt in smaller bowl. Then whisk in sparkling water until combined well.
Working in batches of 3, coat blossoms in batter, lifting each out by its stem and letting excess drip off, then fry, turning, until batter is crisp (it will not brown), 1 to 1 1/2 minutes. Transfer blossoms as fried with a slotted spoon to paper towels, drain, then season with salt.
Drizzle vinaigrette over squash slices, then arrange 3 fried blossoms in middle of each plate. Sprinkle remaining oven-dried tomatoes, pumpkin seeds, and basil around blossoms and season with salt and pepper.