Cut turkey into 1 1/2 inch pieces and place in a 2-3 quart pan. Cook, covered, on medium heat to draw out juices, 8-10 minutes. Uncover, cook on high heat, stirring often, until liquid has evaporated and meat is well browned, 10-12 minutes. Add broth, garlic, bay leaf and oregano, stir to scrape browned bits free from pan. Bring to a boil, then reduce heat, cover and simmer until meat is very tender when pierced, about 1 hour. Uncover, boil on high heat until all liquid has evaporated, 5-10 minutes. Add chilies and vinegar; stir to scrape browned bits free. Remove from heat. Shred meat, using 2 forks or let cool and use your fingers.
Spread beans into a 10" round on a large ovenproof platter. Top evenly with shredded turkey. If made ahead, cover and chill up until next day. Bake, covered, in a 400 oven until hot in the center, 15-25 minutes. Uncover and sprinkle with cheese. Bake, uncovered, until cheese melts, about 5 minutes.
Remove from oven. Tuck 1/3 of the tortilla chips around edges. Accompany with remaining chips and pico de gallo. Scoop layered mixture onto chips to eat.
Beans: In a 10-12 inch frying pan on medium-high heat, stir 1 tablespoon olive oil and 1 large (about 1/2 pound) onion, chopped, until onion begins to brown, 8 to 10 minutes. Add 3/4 cup canned diced green chilies and 2 cans cannellini (white kidney) beans,drained. Stir often, using back of spoon or potato masher to coarsely mash beans, until mixture thickens slightly, about 5 minutes.