Preheat broiler and place rack about 6 inches from heat source. Line a broiling pan with foil. Combine tomato paste, chili powder and a pinch of salt in a small bowl. Rub tomato mixture onto steak, using fingers to press as much as possible onto meat. Place steak on pan and broil for about 6 minutes per side for medium-rare. Remove steak to a cutting board and let stand for 10 minutes. Working against grain, cut steak into very thin slices.
Heat tortillas according to package directions. Lay tortillas flat on a work surface and spread 1/4 cup salsa evenly over bottom of each. Arrange an equal amount of steak on each tortilla, placing it on salsa. Sprinkle 1/4 cup cheese and 1/4 cup shredded lettuce over each serving of steak.
Carefully roll each tortilla into a thick cylinder, tucking in sides as you go. Cut in half crosswise and secure with toothpicks. Serve with extra salsa, chips and avocado slices, if desired.