Preheat the oven to 350 degrees F. Spray a 9-inch square baking pan with nonstick cooking spray and line it with parchment.
For the crumb topping: Combine the melted butter, brown sugar, vanilla, salt and cinnamon in a medium bowl. Stir in the all-purpose flour just until the mixture is crumbly (some dryer bits may remain--that's OK). Toss the mini chocolate chips with the crumb topping and set aside.
For the cake: Combine the eggs, buttermilk, oil and vanilla in a liquid measuring cup and whisk to break up the eggs. Set aside.
Combine the cake flour, granulated sugar, baking powder and salt in the bowl of a stand mixer fitted with the paddle attachment and whisk by hand or mix on low speed until combined, about 30 seconds.
With the mixer running on low speed, add the cubed butter and mix until all of the butter is incorporated into the dry ingredients and the mixture resembles breadcrumbs. With the mixer still running on low speed, add the egg mixture and mix until moistened. Stop and scrape the bowl and paddle, then mix on medium-high speed for 30 seconds.
Pour the batter into the prepared pan and sprinkle 2 tablespoons of the mini chocolate chips over the top. Scatter the crumb topping mixture evenly over the cake batter, then scatter the remaining 1 tablespoon of the chips over top.
Place the baking pan on a rimmed baking sheet to prevent over-browning. Bake until a toothpick inserted in the center of the cake comes out with a few moist crumbs, 43 to 46 minutes. An easy way to test crumb cake is to nudge some of the crumbs away with the toothpick then insert the toothpick into the cake below. Let cool in the pan until barely warm before slicing.
Before cutting to serve, I like to slice off the baked sides of the crumb cake with a serrated knife to reveal the layers of cake and crumb topping. The trimmed bits become a chef's treat.