Bring water to a simmer. Add crushed guayabitas, star anise, and cinnamon. Remove from heat and allow to steep.
To make the melado, combine papelón (or dark brown sugar) with 1/4 cup of water in a small saucepan. Set over medium heat and bring to a simmer. Continue to cook for 5 minutes and set aside to cool.
Combine cornmeal, salt and anise seeds in a large bowl. Strain spice infusion into bowl and discard spices.
Add grated cheese and cooled melado. Knead well. Add warm water one tablespoon at a time as needed to form a smooth mass.
Cover with a damp towel and set aside to rest, 20-30 minutes.
In a deep fryer or heavy pot (3 1/2-4 quarts wide), heat oil to 375 degrees.
Using a silpat or two sheets of wax paper, shape roll out the dough to about 1/4" thick.
Using a round cookie cutter punch out circles about 2 inches wide. The scraps can be gathered together and re-rolled once.
Carefully add the arepitas to the oil, 2-3 at a time and fry until golden about 2 minutes, turning once.
Drain directly on cooling rack or on paper towels. Serve immediately.