Combine first 6 ingredients in a small bowl; set aside.

Combine melon balls in a large bowl, and add lime juice mixture. Cover and chill at least 1 hour. Garnish with mint sprigs, if desired.

Prepare grill.

Heat a medium nonstick skillet coated with cooking spray over medium heat. Add onion; cover and cook 5 minutes, stirring occasionally. Add sugar, 1/4 teaspoon salt, and 1/4 teaspoon pepper; cook, uncovered, 5 minutes or until tender, stirring frequently. Remove onion from pan, and keep warm.

Heat pan coated with cooking spray over medium-high heat. Add mushrooms and 1/4 teaspoon salt; saute 5 minutes or until tender.

Combine 1/4 teaspoon salt, 1/4 teaspoon pepper, Worcestershire sauce, and beef. Divide mixture into 4 equal portions, shaping each into a 1/2-inch-thick patty.

Preheat broiler.

Place patties on a grill rack coated with cooking spray; grill 5 minutes on each side or until done. Place bread on grill rack; grill 1 minute on each side or until toasted.

Arrange bread on a baking sheet. Top each bread slice with 1 patty, 1/4 cup onion, 1/4 cup mushrooms, and 2 tablespoons cheese; broil 2 minutes or until cheese melts.

Combine the first 4 ingredients in a medium bowl. Combine yogurt and remaining ingredients, stirring well with a whisk. Pour over cabbage mixture; toss well. Cover and chill 2 hours.

Preheat oven to 450°.

To prepare crust, lightly spoon 2 cups flour into dry measuring cups; level with a knife. Combine 1/2 cup flour, ice water, and vinegar, stirring with a whisk until well blended to form a slurry. Combine remaining 1 1/2 cups flour, powdered sugar, and 1/2 teaspoon salt in a large bowl; cut in shortening with a pastry blender or 2 knives until mixture resembles coarse meal. Add slurry; toss with a fork until flour mixture is moist.

Divide dough in half. Gently press each half into a 4-inch circle on 2 sheets of overlapping heavy-duty plastic wrap; cover with 2 additional sheets of overlapping plastic wrap. Roll 1 dough half, still covered, into a 12-inch circle. Roll other dough half, still covered, into an 11-inch circle. Chill dough 10 minutes or until plastic wrap can be easily removed.

To prepare filling, combine the apples and lemon juice in a large bowl. Combine 2/3 cup sugar, 3 tablespoons flour, cinnamon, nutmeg, and 1/8 teaspoon salt in a small bowl. Sprinkle sugar mixture over apples; toss well to coat.

Remove top 2 sheets of plastic wrap from 12-inch dough circle; fit dough, plastic wrap side up, into a 9-inch deep-dish pie plate coated with cooking spray, allowing dough to extend over edge. Remove remaining plastic wrap. Spoon filling into dough; brush edges of dough lightly with water.

Remove top 2 sheets of plastic wrap from 11-inch dough circle; place, plastic wrap side up, overfilling. Remove remaining plastic wrap. Press edges of dough together. Fold edges under, and flute. Cut 4 (1-inch) slits into top of pastry using a sharp knife. Brush top and edges of pie with egg white; sprinkle with 1 tablespoon sugar.

Place pie on a baking sheet; bake at 450° for 15 minutes. Reduce oven temperature to 350° (do not remove pie from oven), and bake an additional 40 minutes or until golden. Cool on a wire rack.