Preheat oven to 350 degrees F. Arrange the phyllo shells on a baking sheet in a single layer. (You may need two baking sheets.)
Add the shredded chicken and cream cheese to a large mixing bowl. Mix until well combined.
Add half the bacon, artichokes, spinach, mayo, gouda, havarti, garlic, onion powder, and cayenne pepper to the chicken and mix until thoroughly combined. Season with salt and pepper to taste.
Fill each phyllo cup with about 2 teaspoons of the chicken mixture. Top each phyllo shell with some bacon.
Bake for 5-8 minutes until the cheese has melted and the phyllo shells are golden.
Transfer the chicken artichoke bites to a serving platter before eating warm.
Notes
You can boil the chicken until cooked through, then cool slightly, and shred. You can also use a rotisserie chicken from the store - just discard the skin and shred the meat.
If you are making this the day before do not stuff the phyllo cups ahead of time. Cover the chicken mixture and place it in the refrigerator until needed. If you want to make all of it in advance you can stuff the phyllo cups and freeze them with the mixture in them. When you are ready to serve bake them for slightly longer (10-15 minutes).