Although apple pies have been eaten since long before the discovery of America, "as American as apple pie" is a common phrase for reasons which are not entirely clear. In fact, I can't think of another pie that seems more American to most of us living here in the United States. I wonder how our friends in Europe feel about this.
In my family "Dutch Apple Pie" was a favorite and made often by my mother. We were living in central Ohio and the local Amish community favored a crumb topping consisting of mixed cinnamon, brown sugar, butter (cut in using a pastry knife) and sometimes oatmeal. It made a course crumb topping which sort of melted together in baking. We heard stories that it was sometimes served with a slice of cheddar cheese on top, although I never witnessed this.
A true Dutch Apple pie is made with a lattice top and is served cold whenever you have it in The Netherlands. So this Amish variant has developed somewhere along the way and been passed down to all of us in the Ohio valley. If you ever get to Amish country do make it a point to dine and enjoy one of their many fine, homemade pies. You can't go wrong and most of the restaurants have a dozen or more pie choices on the menu daily.
I frequently make fruit pies, and most often make my apple pies with this crumb topping. It's almost like an apple crisp, but with a bottom crust. I also serve it with a scoop of vanilla bean ice cream and prefer the pie just cooled from the oven or at room temperature rather than chilled.
Use the Cook's Illustrated preferred method (two kinds of apples, tart and slightly sweet) and pre-cook the apples for 10 minutes or more in the microwave. This actually helps the apples set so they will not become mushy and overcooked when you bake the pie in the oven. And do as the Amish do and sprinkle a bit of cinnamon and even some raisins or dried sweetened cranberries into the mix with the apples just prior to baking. Yum!