Cut apple into fourths. Cut core away from each slice. Grate 1 slice into a small bowl; add mayonnaise, granulated sugar, and vinegar, and stir until blended. Season with salt to taste. Finely chop remaining 3 apple slices.
Melt butter with oil in a 10-inch skillet over medium-high heat. Add chopped apples, 1/2 tsp. fennel seed, and crushed red pepper, and cook, stirring often, 10 to 12 minutes or until apples turn golden brown. Reduce heat to medium low, and add brown sugar, whiskey, vinegar, and 1/4 tsp. salt. Cook 2 to 3 minutes or until mixture is thickened. (Mixture will be a syrup consistency.) Remove from heat, and keep warm.
Combine 1/3 cup boiling water, next 3 ingredients, remaining 1/2 tsp. fennel seed, and remaining 1/2 tsp. salt in a 1-cup glass measuring cup. Let stand 10 to 15 minutes or to room temperature.
Preheat grill to 350° to 400° (medium-high). Combine pork and water mixture in a large bowl until blended and liquid is absorbed. (Do not overwork meat mixture.) Shape mixture into 5 thin patties. (Patties should be no more than 1/2-inch thick.)
Grill patties, covered with grill lid, over 350° to 400° (medium-high) heat 4 to 5 minutes on each side or until done. Top each patty with 1 cheese slice during last 1 minute of grilling. Remove from grill, and let stand 5 to 10 minutes.
Spread mayonnaise mixture evenly on cut sides of bottom hamburger buns. Top with burgers. Spoon apple mixture evenly over burgers; cover with tops of buns.