1. Pour olive oil into a 6- to 8-quart ovenproof pan over medium-high heat; when hot, add onion, carrots, and celery and stir often until onion is limp, 6 to 8 minutes. Add 3 quarts water and the beef, peppercorns, and 2 teaspoons salt. Cover pan, increase heat to high, and bring mixture to a boil.
2. Transfer pan, tightly covered, to a 300° regular or convection oven. Bake, turning meat over after 2 hours, until very tender when pierced, about 3 1/2 hours.
3. Lift beef from broth and set on a large, rimmed dish. If meat is tied, cut and remove strings. With two forks, tear meat into large shreds, discarding any large pockets of fat. Skim and discard fat from juices in pan (see notes); add salt to taste.
4. Brush chili oil lightly over cut sides of rolls to taste. Mound meat on roll bottoms and drizzle each sandwich with 2 to 3 teaspoons salsa verde; offer remaining sauce to add to taste. Add salt to taste. Ladle broth into small bowls to dip sandwiches into.