Peel, core, and slice the apples, dropping the slices into a bowl that is holding the 2 quarts of cold water and the juice of the 1/2 lemon. When the apples are sliced, stir them with your hand so that all the slices get coated with the lemon water. Then, drain the apples completely and place them in a bowl that has a cover and pour in 3/4 cup of the sugar, 1 tablespoon (3 teaspoons) of the cinnamon, and, 1/2 teaspoon of the allspice. Stir gently and allow this mix to marinate (covered) in the refrigerator for 24-48 hours.
Once the apples have marinated and you are ready to bake your mini-pies, preheat your oven to 350-degrees F. (If you are having a dinner party, simply prepare the mini-pies in advance, ready to bake, by having them on the baking tray in the refrigerator, covered with cling wrap -- remove the cling wrap and stick them in the oven about 40 minutes before you intend to serve them).
Lay out some parchment paper, or otherwise prepare a flat surface for rolling out the individual biscuits. Dust the surface with some of the flour and flip a biscuit on it to lightly coat both sides and roll it out with a rolling pin. (The kids LOVE to do this!). Add just a little additional flour to each side as needed to keep the dough from sticking to the rolling pin. (I like to rub flour on to my rolling pin before I start, and as needed thereafter). Continue to roll each biscuit until they are each about eight inches across. THEY DO NOT HAVE TO END UP BEING ROUND, OR PERFECT IN ANY WAY! As each crust is finished, stack them on a plate until they're all finished.
On a work surface (I use a cookie sheet, but not the one I'll be baking on), lay out a crust and brush the edges with a hand-beaten mixture of the 2 tablespoons of warm water and the egg white. (Save the remaining egg white blend -- you'll need it again!) Then, after stirring the apples but NOT using the juice in the bottom of the bowl, place about 1/3 to 1/2 cup of the sliced apples on the center of the crust. Spoon on 1 1/2 teaspoons sugar over these apples and dot the apples with a little butter, about a teaspoonfull for each mini-pie.
As each mini-pie crust is filled with the apple blend, fold it over, pressing together the egg-brushed edges so that they stick together. Do this any way you can to get it to stick -- it doesn't have to be pretty!
Spray a large baking tray (I use a commercial half-sheet) with the cooking spray and lay out the closed pies on it so that they are not touching one another too much (or, at all if you can manage this). Then, poke each pie top 2 or 3 times with a fork as to allow steam to escape.
Blend the topping by mixing 2 tablespoons of sugar, 1 teaspoon of cinnamon, and 1/4 teaspoon of ground allspice in a small bowl. Mix these dry ingredients with a fork. Brush the tops of the mini-pies with the remaining egg white and then sprinkle on a good bit of the topping blend.
Bake the mini-pies at 350-degrees F. for 30 minutes. Some of the butter will leak out onto the baking tray but this is normal. As soon as the mini-pies have baked for 30 minutes, remove them from the oven and use a sharp-edged spatula right away to transfer them either to a platter, or, to individual serving plates. Serve hot or very warm.
OPTIONAL: A couple of scoops of good vanilla ice cream really gilds the lily on this sweet dessert!