Preheat the oven to 350°F. Line a 8-inch square baking dish with parchment paper or spray with nonstick cooking spray.
Blend the freeze dried strawberries: In a small, high powered blender, add the freeze dried strawberries and blend on high until they are a fine powder similar to flour, 30 seconds. Set aside.
Mix the wet ingredients: In a large bowl, add the sugar, eggs, Greek yogurt, olive oil, milk, and vanilla and almond extracts and whisk together until well combined and smooth, about 1 minute.
Add the dry ingredients: In the same bowl as the wet ingredients, add half of the strawberry powder (28 grams or about ¼ cup), almond flour, oat flour, baking powder, baking soda, and salt. Whisk until evenly incorporated into the wet ingredients, 1 minute.
Bake the cake: Pour the batter into the prepared pan and smooth the top. Bake until a tester comes out almost clean with just a few crumbs attached, 35 to 45 minutes. Allow the cake to cool completely before frosting.
Make the frosting: In the bowl of a stand mixer fitted with the whisk attachment or in a large mixing bowl (if using a hand mixer), beat the butter on high speed until soft and fluffy, 1 minute. Add the remaining strawberry powder (28 grams or about ¼ cup) powdered sugar, vanilla extract, and milk, and beat on low speed for 30 seconds, then increase to high speed and beat until light and fluffy, about 1 minute.
Frost the cake: Add frosting to the top of the cake and evenly spread it to the edges. Cut into 16 slices. Enjoy as is or serve with fresh strawberries on top or on the side, or top with a few extra freeze dried strawberries.
To store: Cover the pan with plastic wrap and foil and store the cake in the fridge for up to 5 days to ensure freshness.