How Do I Use Ginger in a Cocktail?

Couple of ways.

The easiest, assuming you know how to boil water, is to make simple syrup (1 part each of sugar and water). Bring it to a boil, make sure sugar is dissolved, toss in some peeled and diced ginger, and let it steep off the heat for say 30 minutes. Then strain.

Or you can juice ginger and add sugar to taste. That is probably how they do it at Sylvain. You can also use crystalized ginger or buy ginger syrup.

Don't get the pickled ginger from jars like you've had at sushi joints. Get it fresh. It looks like this

I would definitely steep slivers of it in a simple syrup like BO recommends. Taste as you go. Keep the ginger taste subtle.

But make clean-up easy on yourself. Rather than boiling water in a pot on a cooktop, just add equal parts in a microwavable glass measuring cup (maybe 2 oz each). Nuke it at 30 second increments (stirring in between) until hot and sugar starts to melt. Once at 30 seconds has been good enough for me.

I'm also fond of a more concentrated simple syrup to get the same sweetness with less water dilution. Try two parts sugar to one part water. When recipes call for simple syrup, add half.

Moscow Mule is one of my favorite drinks...

It can be made in a variety of ways, but a recipe that contains fresh ginger is the best.

Marinade shrimp in sake, ginger, garlic, and a touch of soy sauce. Add the shrimp garnish to your favorite bloody mary. Enjoy.