Place chicken breasts in a saucepan with chicken broth, smoked paprika, garlic powder, and black pepper. Bring to a boil, then reduce heat, cover, and simmer for 15-20 minutes, or until chicken is cooked through and easily shreddable. Alternatively, cook in an Instant Pot for 8-10 minutes on high pressure.

Remove chicken from broth (reserve broth for other uses if desired) and shred using two forks, or an electric mixer for quick shredding.

In a medium bowl, combine the shredded chicken with 1 cup of smokey BBQ sauce. Stir well to coat evenly. Set aside.

In a large mixing bowl, combine softened cream cheese, sour cream, mayonnaise, red onion, cilantro, Worcestershire sauce, and hot sauce (if using). Mix until smooth and well combined. Season with salt and pepper to taste.

Add the BBQ-coated shredded chicken to the cream cheese mixture. Stir gently to combine everything.

Fold in 3/4 cup of the shredded cheddar cheese, 3/4 cup of the mozzarella cheese, and 1/4 cup of the Monterey Jack cheese into the chicken and cream cheese mixture.

Transfer the entire mixture to a 9x13 inch baking dish, spreading it evenly.

Top the dip with the remaining 1/4 cup shredded cheddar, 1/4 cup mozzarella, and 1/4 cup Monterey Jack cheeses.

Preheat oven to 375°F (190°C). Bake for 20-25 minutes, or until the dip is bubbly around the edges and the cheese on top is melted and lightly golden.

Remove from oven. Drizzle with a little extra BBQ sauce if desired, and sprinkle with additional fresh cilantro. Serve hot with tortilla chips, crackers, pita bread, or vegetable sticks.