Melt butter over low heat; set aside.
Peel, core, and cut apples into 1/3-inch-thick slices. Place apples in a 9- by 13-inch baking pan (see Notes above); spread level.
In a bowl, mix flour, sugar, cinnamon, baking powder, and salt. Drop in eggs and mix with a pastry blender or fork until crumbly (the mixture will resemble streusel). Spread topping evenly over apples. Drizzle with melted butter.
Prepare your grill for indirect heat. The temperature inside your grill should be between 350° and 400° (insert a long-stemmed thermometer through lid vent to measure temperature). Place the apple crisp on the grill (not directly over the coals or burners). Cover barbecue.
Cook until the topping is browned and the apples are bubbling, 40 to 45 minutes. Serve warm with vanilla ice cream, if desired.
How to set up your grill for indirect heat.
On a charcoal barbecue: Ignite about 60 briquets on the firegrate of your barbecue. Let burn until coals are dotted with ash, about 25 minutes, then push coals to one side and lay a drip pan on the other side. Set grill in place.
On a gas barbecue: Turn heat to high and close lid for at least 10 minutes. Adjust heat to medium and turn off burners as needed to create a hot and a cool zone for indirect cooking. Set drip pan in cool zone; set grill in place.
Smokin' Apple Crisp: If you really love the flavor of smoke, you can actually smoke this apple crisp: Cover 1/3 cup applewood chips with water and soak for about 30 minutes, then drain. Just before you place the apple crisp on the grate, scatter the chips over the coals (if using a charcoal grill), or place the chips in the metal smoking box of your gas grill (or in a foil pan directly on the heat in a corner).