Add the chicken, broth, 1/4 onion, garlic clove, and salt to a pot and bring to a boil and cook at a boil for 20 minutes. Place chicken on a plate to cool and let sit until cool enough to handle then shred with two forks. Reserve some of the broth for the sauce.
While chicken cooks, prepare the chilis verde sauce: place whole tomatillos and pepper in a small pot and cover with water and bring to a boil. Cook until the tomatillos have transformed from their bright green raw state to a dull green, about 10 minutes. Drain water off and transfer to a blender. Add 1/4 onion and the other garlic clove and salt. Process until completely smooth, adding the reserved chicken stock to reach your desired texture.
Preheat the oven to 350ºF and grease an 8x8 inch glass baking dish.
Combine the shredded chicken, Mexican cheese, Monterey Jack, and sour cream in a large bowl. Stir to combine and season to taste with salt and pepper.
Fill the turkey breast slices with the shredded chicken mixture then roll and place seam side down in the greased baking dish. Pour the chili sauce over top, sprinkle with additional cheese, if desired and bake at 350ºF for 20 minutes, or until browned and bubbling. You can also broil for a couple of minutes at the end to ensure a browned topping.
Let cool for 5-10 minutes then plate. Garnish with sour cream, queso fresco, fresh cilantro, diced avocado, and a squeeze of lime juice.