In a 4- to 5-quart (3.8- to 5-liter) pan, bring 8 cups (1.9 liters) water to a boil over medium-high heat. Stir in broccoli and cook until just tender to bite (about 7 minutes). Immediately drain broccoli, immerse in ice water until cool, and drain again. In a food processor or blender, combine a third of the broccoli with garlic, cheese, olive oil, sesame oil, salt, and 3 tablespoons (45 ml) water. Whirl until smooth. Scrape down sides of container, add half the remaining broccoli, and whirl until smooth again. Add remaining broccoli; whirl until smooth. Set aside.
In a 5- to 6-quart (5- to 6-liter) pan, bring about 3 quarts (2.8 liters) water to a boil over medium-high heat; stir in pasta and cook until just tender to bite, 8 to 10 minutes. (Or cook pasta according to package directions.)
Drain pasta well. Transfer to a large serving bowl and stir in vinegar. Add pesto and mix gently but thoroughly. Garnish with tomato and serve immediately.