Place the whole plantains in a large pot filled with boiling water. Simmer, partially covered, for 20 to 30 minutes, until they are thoroughly cooked. Remove from the water and let cool. Peel the plantains, slice, and place in a food processor along with the sugar. Process until smooth. If the dough seems to soft and runny, you may add some all purpose flour to thicken it a bit. You may also let it sit in the refrigerator, covered, for a half an hour so it will harden too.
Make round balls of about 1 to 2″. Press in between plastic rounds in a tortilla press or roll with a roller until you get a flat disk of about ¼”. Place about a tablespoon of refried beans right in the center and fold like a turnover. Press down along the edges so they will be tightly sealed.
In a large deep skillet set over medium heat, heat enough oil to have about an inch high. About 3 to 4 minutes later, when it is hot but not smoking, insert the quesadillas a few at a time. The oil should be bubbling around the quesadillas as you do. Let them fry, about 2 minutes on each side until nicely tanned and lightly crisped. Don’t let them brown too much. Remove them and place on a paper towel to drain excess of oil.