Fill a large pot with water and bring up to a boil over high heat. Add some salt and drop the broccoli. Cook until tender, about 3-4 minutes.
While broccoli is cooking, sauté some onions with 1 tablespoon of EVOO in a skillet over medium-high heat until soft. Add the black beans, tomatillo salsa and season with salt and ground black pepper.
Once the broccoli is done blanching, drain it and toss it into the mixture of onions and beans.
While the onion-bean mixture is cooking, bring the stock up to a boil. Once boiling, remove from heat and stir in the cheese.
Arrange a layer of baked pita chips down on a platter and top with broccoli mixture. Drizzle the nachos with cheddar cheese sauce and serve.