Place a medium pot over medium heat. Add the cream cheese, sour cream, and unsalted butter; whisk or stir until the mixture is smooth and combined.
Stir in the shredded Parmesan and 1 1/2 cups shredded mild cheddar until the cheeses melt and the mixture begins to bubble.
Add the drained chicken, diced jalapeños, rinsed black beans, drained corn, and pressed garlic; stir until evenly distributed and heated through.
Bring the dip to a gentle boil, then remove the pot from the heat.
Transfer the dip to a serving dish if desired, top with additional shredded cheddar, then garnish with diced Roma tomato and diced avocado; serve warm with tortilla chips or crackers.