Position knife blade in miniature food processor bowl; add first 9 ingredients. Process until smooth, scraping sides of processor bowl occasionally.
Spread basil mixture evenly on 4 bagel halves. Top each bagel half with a tomato slice, and sprinkle evenly with cheese.
Place bagel halves on a baking sheet. Broil 5 1/2 inches from heat (with electric oven door partially opened) 1 minute or until cheese melts.