Fill a large pot with water, and season with kosher salt. Bring to a boil. Meanwhile, fill a bowl with ice and water, and set aside.

Cook okra in boiling water 30 seconds; remove and immediately plunge into ice water to stop the cooking process. Repeat with long beans. Add squash to boiling water, and cook until a fork can easily pierce flesh, about 6 minutes. Remove and let cool on a baking sheet about 15 minutes.

Whisk together fish sauce, vinegar, sugar, ginger, and garlic in a bowl. Slowly drizzle in peanut oil while continuing to whisk.

Open bottom and top vents of a charcoal grill completely. Light a charcoal chimney starter filled with briquettes. When briquettes are covered with gray ash, pour them in an even layer on bottom grate of grill. Place tomatoes on oiled grate, and grill, turning often, until completely charred, about 10 minutes. Remove from grill. Place okra, beans, and squash on grate, and grill, turning often, until slightly charred on all sides, 3 to 4 minutes. Cut okra diagonally in half, cut beans into 2-inch pieces, and cut squash into cubes. Gently toss all vegetables in a large bowl; transfer to a serving plate.

Roughly chop tomatoes, and place over vegetables. Spoon dressing over dish. Garnish with chayote shoots and leaves. Drizzle with extra-virgin olive oil, and finish with cracked black pepper.