Preheat oven to 350 degrees. Line a baking sheet with parchment paper or a silicone baking sheet and set aside.
In a large bowl, use a hand-held mixer to beat together the sugar and butter until creamy.
Add the lemon juice, lemon zest, and molasses. Beat into the sugar and butter mixture.
Add the flour, salt, and baking powder. Use a wooden spoon or spatula to fold the dry ingredients into the wet ingredients until just mixed. Do not over mix.
Use a small ice-cream scoop to portion out the cookie batter. Roll into balls and roll the balls into the course white sugar. Place the cookie on the lined baking sheet and press your thumb into the top of each cookie ball to create a dimple.
Bake for 13 minutes. Remove from oven and allow to cool on a cooling rack.
Whisk together the confectioner's sugar and lemon juice. Use a teaspoon to lightly drizzle the glaze over the top of the cookies. The dimple in each cookie will likely collect the glaze, so don't over fill it.
Next, top with a little lemon zest and allow the glaze to harden.
Store in a food-safe container with a lid at room temperature.