Preheat the grill to medium-high heat, 400 degrees F (205 degrees C).
Layout a piece of the foil and create a layer of the thinly sliced lemons.
Lay the snapper on the lemons.
Insert 2/3 of the garlic, chives, and butter into the cavity of the fish.
Cut 5 large grooves into each side of the snapper. Stuff the remaining third of the garlic and chives into the slices.
Top the fish with the remaining lemon slices and butter. Sprinkle with half the chives and drizzle with olive oil then wrap the fish tightly in the foil.
Place the foil-wrapped snapper on the preheated grill and cook for 1 hour.
Unwrap the snapper and cook for an additional 10 minutes or until it gets a nice brown color.
White Wine Sauce
Melt the butter in a small saucepan over medium-low heat. Add the garlic and green onions and cook until transparent, about 4 minutes. Stir in the tarragon, salt, and pepper.
Increase the heat to medium and stir in the white wine and lemon juice. Bring to a simmer for 1 minute. Remove from the heat and stir in the parsley.
To Serve
On a platter arrange the foil baked snapper then drizzle with some of the white wine sauce. Sprinkle with the remaining fresh chives. Place the remaining white wine sauce in a jug so people can add it as desired.