Add the strawberries, lemon juice, and sugar to a small saucepan. Place over medium heat and cover. Let the strawberries heat and begin to break down, about 7 minutes.
Once there's a good amount of liquid in the saucepan from the strawberries, add the cornstarch and mix in well to make sure there are no clumps.
Bring to boil, then turn the heat down to medium-low and let heat, covered, until the strawberries are thick and almost jelly-like consistency.
Remove from heat and let cool.
To make the crust:
Preheat the oven to 375ºF.
Mix together the flax and water and let sit until slightly thickened, about 5-7 minutes.
Sift together the dry ingredients (flour, baking powder, and salt), then add the sugar and mix well.
Cut the butter into small cubes and add to the flour, mixing well to incorporate it until the mixture resembles coarse crumbs. I always do this step by hand, but you can use a pastry cutter or two forks if you prefer.
When the butter is almost fully incorporated, add the flax egg.
Press about 2/3 of the mixture into a parchment-lined 8"x8" square baking pan, and press it down firmly into an even layer.
Pour the strawberry mixture over top of the crust. It's ok if it's still a little warm!
Add the shredded coconut to the remaining crumb mixture, mix well, and sprinkle over the strawberry layer.
Bake 30-35 minutes until golden brown along the edges.
Before pouring the strawberry mixture over the crust, I used a fork to mash the strawberries so it more resembled a jelly, but that's totally optional.