Make the crumb topping: In a medium bowl whisk the 1/3 cup granulated sugar, 1/3 cup dark brown sugar, 1/2 teaspoon ground cinnamon, 1/4 teaspoon ground ginger, and a pinch of salt into the 8 tablespoons melted (still warm) butter until combined. Add 1 3/4 cups cake flour and stir with a rubber spatula or wooden spoon until the mixture forms a thick, cohesive dough. Set the bowl aside to cool to room temperature, at least 10–15 minutes.
Prep the rhubarb: In a small bowl toss the 1 1/2 cups small-diced rhubarb with the 2 tablespoons granulated sugar, 1 teaspoon cornstarch, and 1/4 teaspoon ground ginger until evenly coated. Set aside.
Preheat and prepare pan: Position a rack in the upper-middle of the oven and preheat to 325°F. Spray an 8-inch square baking dish with nonstick spray and line it with parchment paper, pressing the paper into the corners and allowing excess to overhang the sides.
Combine dry cake ingredients: In the bowl of an electric mixer (or a large bowl if mixing by hand), combine 1 1/4 cups cake flour, 1/2 cup granulated sugar, 1/4 teaspoon baking soda, and 1/4 teaspoon table salt. Mix briefly on low speed just to combine.
Cut in the butter: With the mixer running on low, add the 6 tablespoons softened (but still cool) butter, one piece at a time. Continue beating on low until the mixture resembles moist crumbs with no visible large butter pieces, about 1–2 minutes.
Finish the batter: Add the 1 egg, 1 egg yolk, 1 teaspoon vanilla extract, and 1/3 cup buttermilk to the crumb-like mixture. Increase speed to medium-high and beat until the batter is light and fluffy, about 1 minute, scraping down the bowl once if needed.
Drain and flour the rhubarb: Use a slotted spoon to lift the rhubarb from its bowl so most of the released juice stays behind. Transfer the drained rhubarb to a clean small bowl and toss it with the 2 teaspoons all-purpose flour to coat evenly.
Fold rhubarb into batter: Add the floured rhubarb to the cake batter and gently fold it in with a rubber spatula until evenly distributed, taking care not to overmix.
Assemble cake and add topping: Transfer the batter to the prepared pan and use a rubber spatula to spread it into an even layer. Break the cooled crumb topping into large pea-sized pieces with your fingers or a spoon and scatter them evenly over the batter, starting at the edges and working toward the center.
Bake and cool: Bake until the crumbs are golden and a wooden skewer inserted into the center of the cake comes out clean, 55–65 minutes. Transfer the pan to a wire rack and cool at least 30 minutes.
Finish and serve: Lift the cake from the pan using the parchment overhang, place on a cutting board, and dust with powdered sugar just before slicing and serving.