Preheat grill. Divide asparagus into 2 bunches. Align each bunch in a flat row with ends at the same level. Thread 2 skewers crosswise through each bunch in lower third of stems, where thickest, to form a solid mat. Thread remaining 2 skewers horizontally through onion slices, 2 per skewer. Brush asparagus, eggplant, and onions with 1 tablespoon regular olive oil and sprinkle with salt and pepper to taste.
In a glass bowl, make vinaigrette by combining the balsamic vinegar, kosher salt, black pepper, garlic, and shallots; mix well. Slowly drizzle in the extra virgin olive oil while whisking.
Grill asparagus, onions, and eggplant uncovered, on a lightly oiled rack over moderate heat. Turn and brush occasionally with regular olive oil, until tender, 8-12 minutes total.
Remove vegetables from skewers and drizzle with vinaigrette. Top with baby spinach and sprinkle with Parmesan shavings.
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