Preheat oven to 350°F (175°C). Line a 9x13 inch baking pan with parchment paper, leaving an overhang on the sides for easy lifting.

In a large mixing bowl, combine the melted butter and 1/2 cup granulated sugar. Whisk until well combined.

In a separate bowl, whisk together the 2 1/4 cups all-purpose flour, baking powder, and salt.

Add the dry ingredients to the wet ingredients, mixing with a fork or your hands until a crumbly mixture forms. It should resemble coarse crumbs with some larger clumps.

Press about two-thirds (roughly 2 1/4 cups) of the crumb mixture evenly into the bottom of the prepared baking pan. Set the remaining crumb mixture aside for the topping.

In a medium saucepan, whisk together the 1 1/2 cups granulated sugar and 1/4 cup all-purpose flour until there are no lumps.

Whisk in the eggs one at a time until fully incorporated.

Stir in the fresh lemon juice and lemon zest.

Cook the mixture over medium-low heat, stirring constantly with a whisk, until it thickens to a pudding-like consistency (about 5-7 minutes). Do not boil. Remove from heat.

Pour the hot lemon filling evenly over the pressed crumb crust in the baking pan.

Crumble the reserved one-third of the crumb mixture evenly over the lemon filling.

Bake for 35-40 minutes, or until the crumb topping is lightly golden brown and the lemon filling is set (it should not jiggle excessively when gently shaken).

Remove the pan from the oven and place it on a wire rack to cool completely. This is crucial for the bars to set properly.

Once completely cool (this can take 2-3 hours at room temperature, or speed it up in the refrigerator), use the parchment paper overhangs to lift the entire slab of bars out of the pan.

Place on a cutting board and cut into 16 squares.

Dust with powdered sugar just before serving, if desired.