Mix egg pasta, wrap in plastic, and let rest for 1 hour.
Cut pasta ball into 4 wedges.
Pull one wedge out and wrap the others up tightly. Work with one at a time to keep your pasta dough from drying out. Using a rolling pin, flatten the dough out until it is about 1/8 inch thick.
Using an old whiskey glass or cookie cutter with a 3 inch diameter, cut your circles. Cut as many as you can, then form your scraps back up into a ball. Roll it back out and you can get a few more.
Slice up some small cheese triangles. I used cheddar for half of them, and Monterey Jack for the other half. Spoon about a half tspn of the BBQ chicken in the center of your circle, and top it with a slice of cheese.
Keep a glass of water close by to dab your fingers in. You wanna lightly wet the outer edge of your circles (like you're about to seal an envelope). Fold them in half and crimp them shut using a fork. When you get them folded and sealed, place them on a clean towel and let them dry while you continue preparing the rest of the mezzelune.
Let all the mezzelune dry for about about 30 minutes
Heat a large pot of boiling water and boil for 2-3 minutes. When they're floating, they're done.
Scoop out the mezzelune (do not strain). Place them all in a covered bowl until ready to serve.