Preheat the oven to 350 degrees F (177 degrees C).
Prep an 8x8 cake pan but greasing and covering it with parchment paper. Set aside.
First make the persimmon filling. Combine 1 ½ cups of hachiya persimmon pulp, ⅓ cup of light brown sugar, 2 tbsp of all purpose flour, 1 tsp of ground cinnamon, and ¼ tsp of salt. Mix and set aside.
Next, make the streusel topping. Combine ⅔ cup of all purpose flour, ½ cup of light brown sugar, 1 tsp of ground cinnamon, and 4 tbsp of unsalted butter. Break the butter apart with forks and mix together with the other ingredients. Once combined, set aside.
Lastly, make the cake batter. In a large bowl, mix together 1 ¾ cup of all purpose flour, 1 tsp of baking powder, ½ tsp of baking soda, and ¼ tsp of salt. Set aside.
In the bowl of a stand mixer, cream together 6 tbsp of unsalted butter, ⅓ cup of granulated sugar, and ¼ cup of light brown sugar. You can also use a bowl and a hand mixer.
Add the egg and 1 tsp of vanilla extract. Whisk until combined. Make sure to scrape down the bowl if using a stand mixer.
Lastly, add ¾ cup of buttermilk to the wet ingredients and mix until combined. The consistency will look curdled, but that's okay.
Combine the dry ingredients from step 5 to the wet ingredients. Mix until just combined.
Now it's time to assemble! Spread half the cake batter into the bottom of the pan. Spoon the persimmon filling over the top. Follow with the second half of the cake batter. It will be more difficult to spread the second half of the cake batter since the filling is loose, but do the best you can. Finally, top the cake with the streusel topping.
Bake for 45-50 minutes until a toothpick comes out clean.
Cool completely before serving.