1Place potatoes and garlic in a bowl and cover with several inches of cool water; let soak for 4 hours or overnight.
2Preheat oven to 400 degrees F (200 degrees C). Drain potatoes and pat dry.
3Combine potatoes and vegetable oil in a bowl and toss to coat. Spread potatoes slightly apart in a single layer on a baking sheet.
4Bake potatoes in the preheated oven until golden brown, about 25 minutes.
5Sprinkle 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.
6Bake nachos in the oven until cheese melts and chicken is hot, 10 to 15 minutes. Top nachos with tomato, salsa, and sour cream.