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1
Combine the bourbon, gin, lime juice, ginger juice, ginger syrup, and honey (heat gently if needed, so that it is not thick) in a pitcher. Slice the cucumber and add half of the slices to the pitcher. Stir to combine. Taste and adjust if needed: it will be very strong (you'll dilute it later), but it should be balanced. Since ginger juice and ginger syrup can have different levels of pungency and sweetness, add a bit more of either if you want more of a kick or more sweetness. It's up to you! I like it barely sweet, with a ginger flavor that is present but not overpowering.
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2
For each drink: place two slices of cucumber in the bottom of a rocks glass, and muddle them well. Fill glass with ice. Pour approx. 3 ounces of the drink mixture on top of the ice, and top off with club soda to taste. Garnish with another cucumber slice if desired.
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3
Alternatively, you can shake each drink and then top off with a bit of club soda in the glass. If you don't want any fizziness, dilute with cold water.
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4
Combine all ingredients in a saucepan, bring to a boil, and then simmer gently until reduced slightly, 20-30 minutes. Let cool. Strain and store for a couple of weeks in the fridge. (This will also make a great ginger soda, ginger lime soda, etc.)