Place a medium pot over medium heat and add the cream cheese, sour cream, and unsalted butter.
Whisk or stir continuously until the cream cheese and butter melt and the mixture is smooth.
Add the shredded Parmesan and 1 1/2 cups shredded mild cheddar and stir until the cheeses melt and the mixture begins to bubble.
Stir in the drained chicken, diced jalapeños, rinsed black beans, drained corn, and pressed garlic until evenly combined.
Bring the dip to a light boil, then remove the pot from heat.
Transfer the dip to a serving bowl, top with additional shredded cheddar, diced roma tomato, and diced avocado, then serve warm with tortilla chips or crackers.