Add the water and cornstarch to a saucepan and whisk until smooth. Add the strawberries, sugar, and lemon juice. Cook over medium heat, stirring often, until the strawberries soften and the mixture thickens.

Take the saucepan off the heat and let the strawberry filling cool completely. Transfer it to a bowl and place it in the refrigerator until it is thick enough to spread.

Preheat the oven to 350°F. Grease two 8-inch cake pans and line the bottoms with parchment paper. Beat the softened butter, oil, and sugar until creamy. Add the eggs one at a time, mixing after each addition. Add the vanilla and mix until combined.

Whisk the flour, baking powder, and salt in a separate bowl. Add the dry ingredients to the butter mixture a little at a time, alternating with the buttermilk. Mix just until the batter is smooth. Avoid over mixing.

Divide the batter evenly between the prepared pans. Bake for about 30–35 minutes. Check the center with a toothpick toward the end of the baking time. The cake is ready when the toothpick comes out clean or with a few moist crumbs.

Whisk the flour and milk together in a saucepan until smooth. Cook over medium heat, stirring often, until the mixture thickens. Set it aside and let it cool completely. Beat the butter and sugar until light and creamy. Gradually add the cooled milk mixture while mixing. Add the vanilla and salt, then mix until the frosting is smooth.

Let the cakes cool in the pans for a few minutes, then transfer them to a wire rack. Allow them to cool completely. Trim the tops if needed. Place one cake layer on a serving plate and spread the strawberry filling over the top. Place the second cake layer over the filling.

Spread the frosting over the top and sides of the cake. Smooth it with a spatula or leave a little texture on the sides. Chill the cake until the filling and frosting have set enough for easy slicing. Cut into 10 pieces and serve.