Preheat the oven to 350 degrees F (175 degrees C).
Combine flour, melted butter, and powdered sugar in a bowl and stir to combine. Pat the dough into the bottom of an 8x8-inch square baking pan using your fingers. Prick the crust all over with a fork to keep it from rising in the oven.
Bake in the preheated oven until crust just starts to brown, 10 to 12 minutes. Remove from the oven and let cool.
Combine eggs, white sugar, lemon juice, lemon zest, and lemon extract in a mixing bowl. Beat until the sugar is dissolved and the mixture is smooth. Whisk flour and baking powder together in a separate bowl, then add to the lemon mixture. Continue to beat until the lemon filling is smooth, about 2 minutes.
Pour the lemon filling on top of the pre-baked pastry crust.
Bake in the preheated oven until the filling is set, 23 to 25 minutes. Remove from the oven and let cool to room temperature, 20 to 30 minutes.
Cut into 6 bars, then dust with powdered sugar. Top each bar with fresh raspberries.