Make the peach puree: In a blender, combine the sliced peaches and 1/2 cup of water. Blend until smooth. Strain through a fine mesh sieve to remove any pulp, if desired.

Create the basil syrup: In a small saucepan, combine 1/2 cup water, sugar, and basil leaves. Heat over medium until sugar dissolves, about 3 minutes. Remove from heat and let steep for 10 minutes. Strain out basil leaves.

Combine ingredients: In a large pitcher, mix the peach puree, basil syrup, lemon juice, and remaining 3 cups of cold water. Stir well.

Chill: Refrigerate the lemonade for at least 1 hour to allow flavors to meld. Alternatively, serve immediately over ice.

Serve: Fill glasses with ice cubes, pour the lemonade, and garnish with lemon slices and fresh basil sprigs.

Adjust sweetness: Taste before serving and add more sugar or lemon juice if needed to balance flavors.