Heat the oven to 325°F and arrange the racks in the upper and lower sections of the oven. Line 2 baking sheets with parchment paper or a silpat and set aside.

Combine the flour, ground ginger, cinnamon, baking soda, and salt in a large bowl and whisk together.

Combine the 3/4 cup sugar and the butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until light and fluffy, about 3 minutes. Scrape down the sides of the bowl and add the eggs and molasses and beat until smooth, about a minute more.

Scrape down the sides of the bowl again and add the flour mixture. Mix on low speed until incorporated, about 1 minute. Wrap half of the dough in plastic wrap and refrigerate. Divide remaining dough into roughly 32 heaping teaspoon sized rounds and place on the prepared baking sheets.

Place the remaining 1/3 cup of sugar in a bowl. Roll each piece of dough into a ball and roll each ball in the sugar to coat. Space the balls about an inch apart on the baking sheets and bake for 12-14 minutes, rotating them from the upper and lower racks halfway through. They will still be soft when they are done.

Immediately transfer cookies to a wire rack and repeat with the remaining dough. Let cookies cool completely before frosting.

Place the powdered sugar, butter, and zest in the bowl of a stand mixer fitted with the paddle attachment and mix on low speed until mixture looks crumbly. Gradually increase the speed to medium and beat until smooth, about 2 minutes. Reduce the mixer to low and add lemon extract and juice and beat until combined. Increase the speed to medium and beat until fluffy.

When cookies are completely cool, spread or pipe about 1 heaping teaspoon of filling onto the flat side of half the cookies. Top with another cookie and press to help them stick together and flatten the frosting.

Cookies keep sealed in an airtight container at room temperature for up to 3 days or freeze for up to 3 months. Defrost (tightly wrapped) at room temperature.