Makes about 4 cups, 6 to 8 servings
Place sugar, thyme, and zest in a food processor or blender and pulse until thyme is finely ground, about 1 minute. Transfer mixture to a medium bowl and add pepper, 2 cups water, lemon juice, vodka, and salt and whisk to combine. Let mixture stand until sugar and salt have dissolved, stirring occasionally, up to 10 minutes.
Pour mixture into a 9-inch square metal baking pan and place it in the freezer. Stir mixture, pulling a fork back and forth through it, every 45 minutes or so for 4 hours or until frozen. The mixture will become fluffy and icy.
Shuck oysters and serve with small scoops of granita spooned on top. Garnish with a slice of jalapeno.