Mix butter and garlic in a skillet over medium-high heat until butter is melted. Add chicken; cook and stir until chicken is heated through, 2 to 4 minutes.
Stir cream cheese, Parmesan salad dressing, and vinegar into chicken mixture; cook until well-blended and warm, about 5 minutes. Mix 1/2 of the Monterey Jack cheese and 1/2 of the Parmesan cheese into chicken mixture; transfer to a slow cooker. Sprinkle remaining Monterey Jack cheese and Parmesan cheese over dip.
Cook on Low until hot and bubbling, 30 to 40 minutes.
Instead of a slow cooker, you can put this into a casserole and cook it in a 350 degrees F (175 degrees C) oven for about 15 to 20 minutes, until cheese on top is golden and bubbly and dip is heated through.
Two 10-ounce cans chicken can be used in place of the shredded chicken.
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