Carefully brush oil on grill rack. Place patties on grill over medium heat. Cover grill; cook 10 to 12 minutes, turning once, until meat thermometer inserted in center of patties reads 160°F. Place cheese slices on patties during last 1 to 3 minutes of cooking time. Add buns, cut sides down, to grill for last 1 to 2 minutes of cooking time, until lightly toasted. Place one patty on each bun bottom. Top each with 2 tablespoons avocado mash and remaining 1 tablespoon salsa; replace tops.