To make the crust; cream butter and sugar until fluffy. Add egg, vanilla extract, and sour cream; mix well.
Sift together flour, baking powder, and salt and add to the butter mixture; mix well.
Turn out the dough on to a working surface and form into a disk. Wrap the dough and chill for 30 minutes before using.
To make the filling; cook the apples in a skillet over medium heat until somewhat soft, about 5 minutes. Add the brown sugar and cinnamon and mix well. Continue to simmer over low heat, stirring occasionally, for 10 minutes or until the apples are soft. Mash the apples coarsely, leaving some small chunks visible. Let it cool slightly.
Preheat the oven to 350˚F. Grease a 9-inch springform pan or a pie dish. Cut out 1/3 of dough and keep it in the fridge. Stretch our the 2/3 of dough with your hand and press down on the bottom of pan and 1-inch up the side of pan as well.
Put apple filling on the bottom crust and spread evenly.
Roll our the remaining 1/3 of dough into 1/8-inch thickness, and cut into 3/4-inch strips. Use the dough strips to create the lattice work on top of apple filling. Using a fork, cramp the strips onto the bottom crust. Sprinkle a little extra sugar on top crust.
Bake the pie in a preheated oven for 45-50 minutes. Let the pie rest in the pan for 10 minutes on the cooling rack, then remove the rings on the side. Cool completely.