Preheat the oven to 350˚F (177˚C). Arrange racks in lower and upper thirds of the oven. Line 2 baking sheets with parchment papers. Set aside.
In a large mixing bowl with an electric mixer fitted with a paddle attachment, beat the butter, both sugars, baking powder, baking soda, and kosher salt on medium speed until light and smooth, about 3 minutes. Add vanilla extract and egg, beat until blended. Add the cocoa powder and mix on low speed until combined.
Add the flour and milk to the wet ingredient alternately in 3 additions, by starting and finishing with the flour. Mix on low speed until just combined, scrapping down the sides and bottom of the bowl every once a while. Do not overmix.
Scoop batter with a cookie scoop (2 tablespoons) onto the prepared baking sheets, leaving about 2-inch spaces in between. You can fit about 8 scoops on one sheet.
Bake in the oven for 11 – 12 minutes, rotating the baking sheets top to bottom and front to back after 6 minutes. Bake until set and toothpick comes out clean. Transfer the baking sheets to counter and let rest on the sheets for at least 5 minutes. Transfer the cakes to cooling racks and cool completely. Repeat with the rest of the cake batter.
For the buttercream, first transfer the freeze-dried strawberries into a food processor (or grinder) and process into fine powders. Measure out 1/2 cup powders to use.
In a large mixing bowl with an electric mixer fitted with a whisk attachment, beat the butter for a minute on medium speed. Add strawberry powders, powdered sugar, rose extract, salt and 3 tablespoons milk. Mix for 30 seconds on low speed, and beat on medium-high speed for 2 minutes. Add more milk if the buttercream is too thick.
On a cool baking sheet, line a parchment paper. Place half of the cakes upside down on the baking sheet.
Prepare a piping bag with a ½-inch round tip. Fill the bag with buttercream. Pipe a dome of frosting onto the center of an upside-down cake. Top with another cake. Gently press down to squeeze the filling to the edge. Repeat with the rest. Serve.