Bring a large pot of salted water to a boil over high heat. Fill a large bowl with ice and water; set aside. Add leeks to boiling water, and press down using tongs to submerge. Return water to a boil over high; cover and reduce heat to medium-low. Simmer, covered, until leeks are just tender when pierced with a paring knife but not falling apart, 3 to 4 minutes. Lift leeks out of pot using tongs, and submerge in ice bath. Let stand until cooled, about 10 minutes.

While leeks cool, pulse pine nuts, 2 tablespoons olive oil, vinegar, garlic, and ½ teaspoon salt in a food processor until nuts are finely chopped, about 8 pulses. Drizzle in ¼ cup tap water, and pulse until mixture forms a loose chunky-peanut-butter consistency, about 5 pulses. (Do not puree until smooth.) Transfer to a small bowl; season with additional vinegar and salt to taste. Cover and set aside.

Preheat oven to broil with oven rack 6 inches from heat. Brush a broiler-safe rimmed baking sheet with 1 tablespoon oil. Pat leeks dry, and arrange, cut side up, on baking sheet. Brush leeks with remaining 2 tablespoons oil. Sprinkle evenly with pepper and remaining ¼ teaspoon salt. Broil leeks until lightly charred, 7 to 10 minutes.

Just before serving, whisk 1 tablespoon boiling water into pine nut sauce to loosen, adding additional water as needed to reach desired consistency. Spread about ½ cup sauce on a platter, and arrange leeks over sauce; spoon remaining sauce over leeks. Garnish with additional toasted pine nuts.