Here, one dough does double the work (way to go, dough!): Make it by rubbing cold butter into the dry ingredients, as you would for pie crust. Reserve a half-cup of this mixture, mix it with a teaspoon of crunchy sugar, and you've cut your crumb top. Boom! Done.
Take the rest of the sandy mess, mix in an egg and sour cream (or sub in another dairy product of similar viscosity—see how here), and you'll get a sticky batter you might have to tussle with—just a little—to spread into and up the sides of a spring-form pan, like a thick shortbread crust (I find that damp hands help). Pour over a blend of fresh strawberries, strawberry jam (to concentrate the fruit flavor), and cornstarch (to stave off wateriness), then drop more of that wet batter over top. On goes the crumble you already made, and your work is over.
In a blender, combine the strawberries and jam and process until combined. Make a slurry of the cornstarch and vanilla extract, add to blender, and blend until smooth. Set aside for later.
For the cake and crumb topping:
Preheat the oven to 375° F and grease a 9-inch springform pan.
In a large bowl, whisk together the sugar, flours, baking powder, and baking soda. Sprinkle in butter cubes and use your fingers to rub them in—as if you were making pie dough—until mixture resembles large coarse crumbs with pea- and walnut-sized butter pieces.
Remove 1/2 cup and set it aside for the topping in a medium bowl.
To the large bowl, add sour cream, egg, and vanilla. Mix well.
Dollop a little over half of this batter into pan. With slightly damp hands, pat the batter across bottom of pan and about 1 inch up sides; the mixture will be very sticky and somewhat uneven.
Pour over the strawberry purée in an even layer over the batter, leaving a rim of dough around it. Cover with blobs of remaining cake mixture, trying to spread them as evenly as possible and using damp hands to thin them out. Don't stress about it! (Trust me.)
Add the Demerara sugar to the reserved 1/2 cup dough and stir with a fork to mix. Sprinkle evenly over cake.
Bake cake until lightly golden, about 45 minutes. Cool completely before serving.