In a medium bowl, stir together all ingredients until combined. Taste and adjust seasonings as needed.
Place on a sheet of plastic wrap and wrap up in a log. Chill in the fridge for at least 1 hour.
To make the potatoes:
Use a sharp knife to cut the potato into thin slices, leaving 1/4 inch at the bottom unsliced so that the potato stays in one piece. If it’s helpful, place 2 pencils or chopsticks on either side of the potato as you cut, which will prevent you from slicing all the way through.
Cut the chilled butter into thin slices and wedge down into the cuts made in the potato.
Tear sheets of aluminum foil into squares and place potatoes into the center of each one.
Sprinkle potatoes generously with coarse salt and wrap tightly in the aluminum foil.
Preheat grill to medium high heat so that there are areas of indirect heat and direct heat. For a charcoal grill, push all the briquettes to one side. For a gas grill, leave half of the burners unlit.
Place potatoes over the indirect heat portion of your grill and cook for about 45 minutes, rotating occasionally, until potatoes are cooked through.
Serve potatoes with additional chive butter.
Note: If using unsalted butter, increase salt in chive butter to 1 teaspoon.