Preheat oven to 350 degrees Fahrenheit. Spray an 8x8 baking pan with cooking spray and line with parchment paper.
In a small bowl, add arrowroot starch and water. Stir to combine. Allow to sit for 2-3 minutes.
In a medium bowl, add cane sugar followed by melted butter, agave, lemon zest, lemon juice, and vanilla extract. Whisk to incorporate. Add arrowroot starch mixture and whisk again.
Add almond flour, coconut flour (make sure you packed your tablespoons), salt, and baking soda. Stir until batter forms. Fold in poppyseeds. Allow to sit and absorb liquids for 2-3 minutes.
Transfer batter to prepared pan.
Using an extra piece of parchment paper, flatten batter (will be dough-like) into an even layer, making sure all corners are filled.
Bake for 16-19 minutes. The edges will be slightly browned and the top a golden brown. Remove from oven and allow to cool completely on wire rack.
While bars are cooling, rinse out and dry bowl, and add frosting ingredients. Using electric mixer, beat until frosting becomes smooth. Store in refrigerator until ready to frost.
When cake is cool, chill in refrigerator for at least 30 minutes.
When ready to frost, using a spatula, add frosting and gently frost cake. Grate additional lemon zest over top if desired. Remove cake from pan using parchment flaps. Slice into squares.
Store leftovers in an airtight container.