Preheat oven to 350°F and line an 8x8-inch square baking pan with parchment paper. A glass or stoneware pan works best.
In a medium glass bowl, add the granulated sugar and lemon zest. Rub the zest into the sugar to release the natural oils. Whisk in the melted butter and powdered sugar.
Next, whisk in the eggs one at a time until each egg is thoroughly combined. Then whisk in the lemon juice, lemon extract, and salt.
Fold in the flour until it’s completely incorporated. Transfer the batter to the prepared pan and spread it evenly over the bottom.
Bake for 20-22 minutes. When done, the edges will be set and the center will still look slightly wet. Cool completely.
Once cooled, make the glaze by mixing all the glaze ingredients, except 1 tablespoon lemon juice, together in a bowl. Add more lemon juice until your glaze reaches your desired consistency. It should be slightly thick but should still run off the spoon.
Pour the glaze over the bars and use an offset spatula to smooth it out in an even layer. Let the icing set/harden before slicing. Store in an airtight container at room temperature for up to 3 days or in the refrigerator up to 5 days.
Notes
Rubbing the zest with the granulated sugar is an important step. It brings out the oil in the zest which lends a stronger lemon flavor to the bars.