1. Prepare the cucumber: Wash the large cucumber thoroughly. If using a regular garden cucumber, you might want to peel it to avoid any bitterness from the skin. For English cucumbers, peeling is usually not necessary. Cut the cucumber into rough chunks that will fit easily into your blender.

2. Blend the cucumber and mint: Place the cucumber chunks and the packed fresh mint leaves into a blender. Add about 1/2 cup of the cold water to help with blending. Blend on high speed until the mixture is smooth and no large pieces remain. You should have a vibrant green liquid.

3. Strain the mixture: Place a fine-mesh sieve over a large pitcher or bowl. Pour the blended cucumber and mint mixture through the sieve, pressing down on the solids with the back of a spoon or a spatula to extract as much liquid as possible. Discard the pulp left in the sieve. This step ensures your lemonade is smooth and free of fibrous bits.

4. Make the simple syrup: In a small saucepan, combine the granulated sugar with 1 cup of cold water. Heat over medium heat, stirring constantly, until the sugar is completely dissolved. This creates a simple syrup that easily incorporates into the cold liquid without leaving grainy sugar at the bottom. Let it cool slightly.

5. Combine all ingredients: To the pitcher containing the strained cucumber-mint liquid, add the fresh lemon juice, the cooled simple syrup, and the remaining 3 cups of cold water. Stir everything together thoroughly until well combined.

6. Taste and adjust: Taste the lemonade and adjust the sweetness or tartness to your liking. If it’s too sweet, add a little more lemon juice. If it’s too tart, add a touch more sugar (dissolve it in a tiny bit of hot water first).

7. Chill and serve: For the best flavor, refrigerate the lemonade for at least 30 minutes to allow the flavors to meld and to ensure it’s perfectly chilled. Serve over plenty of ice in glasses, garnished with fresh lemon slices and extra mint sprigs for an appealing presentation.