Preheat oven to 350°F (180°C) with the rack in the center position. Grease a 9-inch springform pan with butter, line the bottom with parchment, then lightly dust the sides with flour and set aside.
In a large bowl, whisk together 2 cups flour, baking powder, and salt until evenly combined.
In a separate large bowl, beat softened butter and 2/3 cup sugar with an electric mixer on medium-high for 3–4 minutes until light, pale, and fluffy. Add the egg and vanilla extract and beat for 1 more minute.
Add the flour mixture to the wet ingredients and mix on low speed just until no dry pockets remain — do not overmix. The dough will be thick and slightly sticky.
Press two-thirds of the dough evenly across the bottom of the prepared pan and about 1 inch up the sides to form a shallow border. Refrigerate while you prepare the filling.
Whisk together applesauce, cornstarch, cinnamon, and nutmeg in a medium saucepan until smooth, then cook over medium heat, stirring constantly, for 3–4 minutes until thickened. Remove from heat, stir in lemon juice, and let cool 5–10 minutes.
Combine 1 1/2 cups flour, 2/3 cup sugar, and 1 tsp cinnamon in a bowl, then cut in the cold butter pieces using a pastry cutter or fingers until coarse crumbs with some pea-sized chunks form. Stir in almonds if using, squeeze into clumps, and refrigerate until needed.
Remove the pan from the refrigerator and spread the cooled applesauce filling evenly over the dough base. Scatter any remaining cake dough over the filling, then crumble the streusel loosely over the entire surface without pressing it down.
Place the pan on a baking sheet and bake for 45–55 minutes until the streusel is golden brown and a toothpick inserted into the cake portion comes out clean. Tent loosely with foil if the streusel browns too quickly after 30 minutes.
Cool in the pan on a wire rack for at least 20 minutes before releasing the springform ring. Allow to cool completely (1–2 hours) before slicing for clean cuts, then serve at room temperature or slightly warm.