Prepare a high gas or charcoal grill fire. In a colander, toss the squash with 2 tsp. kosher salt and drain for 30 minutes; transfer to a large bowl.
Meanwhile, put the the basil, Parmigiano-Reggiano, 1/4 cup of the olive oil, and 1/2 tsp. salt in a food processor and purée until smooth.
In a small saucepan over medium-low heat, boil the balsamic vinegar until syrupy and reduced to about 2 Tbs., 8 to 10 minutes.
Toss the squash with the remaining 2 Tbs. extra-virgin olive oil and a few grinds of
freshly ground black pepper. Grill, flipping once, until golden and
tender, 8 to 12 minutes. Arrange on a platter, dot with the pesto, and
drizzle with extra-virgin olive oil and the balsamic syrup to taste. Sprinkle with pine nuts and serve.