Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch cake pan.
Combine flour, butter, 1 1/2 egg, 1/3 cup plus 1 tablespoon sugar, and 1 teaspoon cinnamon in a large bowl. Knead by hand to form a smooth, consistent dough.
Turn dough out onto a lightly floured work surface. Roll out 3/4 of the dough into a 10-inch circle; press over the bottom and sides of the prepared cake pan.
Mix apples, sultanas, 1/4 cup white sugar, and 1 tablespoon cinnamon together in a bowl. Spread over the dough in the cake pan.
Roll out remaining dough into a circle; cut into strips about 1/3-inch wide. Form an X in the center of the tart with the 2 longest strips. Create a lattice by weaving vertical and horizontal strips in an over-and-under pattern, using the shortest strips for the edges. Press ends of the strips firmly to the edge of the tart; trim excess dough with a knife.
Brush remaining beaten egg over the dough.
Bake in the preheated oven until pastry is light brown, 60 to 65 minutes.