Place potatoes in a bowl of cold water for 20 minutes.
Meanwhile, make the cheese sauce. Melt butter in medium saucepan. Add flour, and stir until combined. Add milk; bring to a simmer; stir and cook until slightly thickened. Add cheese, cumin, and chili powder. Stir together and cook over medium-low heat until completely melted.
Drain and pat dry potato slices. Heat an inch or two of oil in a large saucepan to 375 degrees. Fry potatoes in batches until lightly browned, about 10 minutes. Drain on a rack or paper.
Once potatoes are cooked and cheese sauce melted, assemble the nachos. Put fried potatoes on a large baking dish. Top nachos with salsa, cheese, and black beans. Place in oven under broiler set on high until bubbling, about 2 minutes. Serve with preferred toppings.