In a large pot, heat the vegetable oil over medium heat to 350 degrees F.
Place 1 sheet of phyllo on a dry work surface. Place one slice of cheese about 5 inches from the top of the narrow end of the sheet. Fold the top of the narrow end over the cheese, leaving 1 inch between the fold and the cheese. Next fold over 1 of the long sides then the other, again leaving 1-inch between the folds and the cheese. Then fold the cheese over, again leaving 1-inch between the fold and the cheese. Continue folding to make a package about 3 1/2 by 4-inches. Continue with the remaining pieces of cheese and phyllo.
Fry the phyllo and cheese packages, 2 or 3 at a time, in the oil until golden, about 2 minutes per side. Drain on a baking sheet lined with paper towels.
Meanwhile, heat the figs and honey in a small saucepan over low heat until the honey is warm. Set aside until the cheese packages have finished frying.
To serve, place 1 cheese package on a plate. Spoon some figs and drizzle some honey over the top of each cheese package. Sprinkle with black sesame seeds and serve immediately.