Preheat oven to 350 °F (177 °C) and butter and flour two 8" round baking pans. Place a 8" round piece of parchment paper into the bottom of each cake pan, as well. Set aside.
In the bowl of a stand mixer, fitted with the paddle attachment, beat butter and sugar until light and fluffy, about 5 minutes, scraping down the sides of the bowl as necessary. In the meantime, whisk together the egg, egg yolks and vanilla extract together in a small bowl. With the mixer on low, gently pour in egg mixture and mix until just incorporated.
In a large bowl, whisk together 3 cups of cake flour, baking powder, baking soda, and salt.
Add 1/3 of the flour mixture to the butter mixture and gently mix until incorporated. Then add half of the milk and mix until incorporated. Add 1/3 of the flour mixture then add the rest of the milk. Mix until incorporated. Add the final 1/3 of the flour mixture and mix until everything is well incorporated.
Using the remaining 2 tbsp. of cake flour, toss the blueberries in the cake flour until they're all coated. Gently fold the flour covered blueberries into the batter with a spatula, careful not to burst them.
Divide batter evenly amongst the prepared baking pans. Bake for 50-60 minutes. Let cool for 15 minutes then invert onto a wire cooling rack to cool completely.
In the meantime, make the lavender simple syrup by combining sugar and water in a small sauce pot and bringing to a gentle simmer until sugar has dissolved. Add the lavender and with the back of your wooden spoon, crush the lavender to release more of the fragrance and flavor. Remove from heat and let steep for 30 minutes.
While the cakes are cooling and the lavender syrup is steeping, make the butter cream by mixing together butter, confectioners sugar, milk in a bowl of a stand mixer. As the mixer is going, on low, throw in the blueberries. They will burst and create a nice light purple color to the buttercream and you'll see flecks of the blueberries in the buttercream. If your buttercream is too thick, add more milk, 1 tsp. at a time. If your buttercream is too runny, add more confectioners sugar.
Once your cake layers have cooled you can begin to assemble them. Gently grab one layer of the cake and place it on your cake stand. Place pieces of wax paper under the cake, all around the cake stand, so when you finish icing the cake, you can remove the wax paper and have a clean cake stand :)
Pour half of the lavender simple syrup on the top of the cake, let soak into the cake for a few minutes.
Take your offset spatula and place a generous amount of buttercream in the middle. Working from the middle out, spread the buttercream all over the layer, leaving a 1" border.
Gently grab the top layer of your cake and place it on top of the bottom layer. Pour the remaining half of the lavender simple syrup on the top of the cake and let soak into the cake for a few minutes. Place a generous amount of buttercream in the middle, and again, work from the middle out and go all down the sides and all over until you cover the entire cake.
Cut and serve! You can keep this cake in an airtight container for up to 5 days. I would keep it in the fridge and let it come to room temperature before eating again.