In a large pot, place the chicken breast and add bay leaves, chili powder, cumin, garlic, salt, pepper, and onion. Pour in enough water to fully submerge the chicken. Bring the water to a gentle simmer over medium-high heat. Once simmering, reduce the heat to low, cover, and cook until the chicken reaches an internal temperature of 165°F, approximately 10-15 minutes.
Remove the pot from the heat and let the chicken rest in the cooking liquid for 3 to 5 minutes. Drain the liquid, then shred the chicken into 1-inch pieces using two forks. Combine the shredded chicken with 1 cup of salsa verde in a large bowl, mixing thoroughly.
Preheat the oven to 400°F with the rack positioned in the center.
Arrange half of the tortilla chips on a rimmed sheet pan. Layer with one third of the shredded chicken mixture and 1 cup of shredded pepper Jack cheese. Repeat with the remaining chips, followed by another cup of cheese and the remaining chicken. Top with the final cup of cheese. Bake for 5-7 minutes, or until the cheese is fully melted and bubbly.
After baking, garnish the nachos with crumbled Cotija cheese, sliced avocado, cilantro, and radishes, if desired. Serve with a dollop of sour cream and the remaining salsa verde for extra flavor.