Butter 2 8-inch cake pans and line with parchment paper.
In a medium bowl, whisk together the flours, baking powder, and salt.
In a large bowl, place the sugar and lemon zest. Using your hands, rub the lemon zest into the sugar for a minute or so until fragrant to release the lemon oil.
Add the butter to the bowl with the sugar and lemon zest. Using a stand or handheld mixer, cream the butter into the sugar-zest mixture, beating on high speed for about 3 minutes until smooth and creamy. Beat in the eggs one at a time, scraping down the bowl as necessary. Beat in the vanilla until well combined.
Add the flour mixture in three additions with the milk: flours, milk, flours, milk, and finish with the flours. Mix well on medium speed to make sure all the flours are incorporated.
Divide the batter evenly between the two pans. Cover and let rest for 10 minutes, then preheat oven to 350° F. Let cake batter continue to rest until oven is ready.
Place the cake in the oven and bake for about 30-35 minutes, checking at 25 minutes, until a tester inserted in the middle comes out clean. Remove from oven and place on a wire rack; let cake cool in pans for 10 minutes, then turn out onto the rack to cool completely.
Make Frosting
With a handheld or stand mixer, beat the butter on medium speed until creamy, about 2 minutes. Add powdered sugar, lemon juice, and zest with the mixer running on low. Add the milk. Increase mixer speed to high speed and beat for 3 minutes until light and fluffy. If you think you need it, add a bit more sugar or milk for desired consistency.
Assemble Cake
Place one cake layer on a plate. Spread a very thin layer of buttercream on the layer, then top with the strawberry jam. I went light on the jam, but you could increase it as you wish.
Top with the other cake layer. Frost the top and sides of the cake with the remaining buttercream.
Store the cake, well-covered, in the fridge, but serve at room temperature for best texture and flavor. Cake will keep in the fridge for 5 days.