Heat the oven to 350 degrees F and lightly grease a 9 x 13-inch baking pan. Whisk the flour, baking powder, and salt together.
Beat the egg yolks with 3/4 cup granulated sugar until pale and thick. Mix in the oat milk and 2 teaspoons vanilla extract, then stir in the flour mixture.
Beat the egg whites until foamy, gradually add the remaining 1/4 cup sugar, and continue beating to stiff peaks. Fold the egg whites gently into the batter in three additions.
Spread the batter in the prepared pan and bake for 24 to 28 minutes. Cool the cake for 20 minutes.
Whisk the evaporated oat milk, condensed coconut milk, full-fat coconut milk, strawberry puree, and 1 teaspoon vanilla extract together until smooth.
Poke the cake all over with a skewer, then slowly pour the strawberry milk mixture across the surface. Cover and refrigerate for at least 6 hours so the cake absorbs the soak.
Spread the coconut whipped topping over the cold cake and arrange the sliced strawberries on top.
Refrigerate the finished cake for 30 minutes, then serve it well chilled.
