Place potatoes and garlic in a bowl and cover with several inches of cool water; let soak for 4 hours or overnight.
Preheat oven to 400 degrees F (200 degrees C). Drain potatoes and pat dry.
Combine potatoes and vegetable oil in a bowl and toss to coat. Spread potatoes slightly apart in a single layer on a baking sheet.
Bake potatoes in the preheated oven until golden brown, about 25 minutes.
Sprinkle salt over potatoes. Spread 1/2 the Cheddar cheese, 1/2 the Monterey Jack cheese, and 1/2 the mozzarella cheese over the potatoes. Sprinkle 1/2 the onions, 1/2 the chicken, and 1/2 the jalapeno peppers respectively over the cheese. Repeat layering with remaining cheeses, onion, chicken, and jalapenos.
Bake nachos in the oven until cheese melts and chicken is hot, 10 to 15 minutes. Top nachos with tomato, salsa, and sour cream.