Preheat the oven to 350°F (175°C). Grease a 9-inch baking pan with nonstick spray and line the bottom with parchment paper, then spray again.
In a medium bowl, mix the flour, baking powder, and salt together, and set aside.
Blend the freeze-dried strawberries in a food processor until they become a fine crumb. Mix them into the dry ingredients.
Pulse the fresh strawberries in the food processor 3-4 times until you have small and medium pieces.
Finely chop the basil and set it aside.
In a large bowl using an electric mixer, beat the sugar, eggs, and lemon zest until the mixture is thick, fluffy, and pale, about 3-4 minutes. Slowly add the olive oil while continuing to beat on medium-high speed. Add the chopped basil and beat for another 2 minutes. Reduce the speed to low, and add the fresh strawberries alternately with the dry ingredients, ending with the dry ingredients.
Pour the batter into the prepared cake pan. Bake for 30 minutes, or until a toothpick inserted into the center comes out clean (a few crumbs are okay).
Cool the cake in the pan on a rack for 10 minutes, then remove from the pan and allow to cool completely on the rack.
Once cooled, dust with powdered sugar or top with mascarpone whipped cream before serving.