Celery Simple Syrup:
2 cups chopped celery
1/2 teaspoon celery seed
1/4 teaspoon salt
1/2 cup white sugar
1 cup ice cubes, or as needed
1 1/2 fluid ounces vodka
3/4 fluid ounce lemon juice
3 pickled jalapeno pepper slices
Place celery, celery seed, and salt in a food processor. Pulse several times until completely blended. Press mixture through a fine mesh strainer with the back of a wooden spoon into a bowl; you will be left with about 1/2 cup of liquid.
Place celery liquid and sugar in a small saucepan over medium heat. Stir until sugar is completely dissolved, about 3 minutes. Remove from heat. Refrigerate until needed.
Place ice, vodka, 3/4 fluid ounce celery simple syrup, lemon juice, and 2 jalapeno slices in a cocktail shaker. Shake for 30 seconds; strain into an ice-filled cocktail tumbler. Garnish with remaining jalapeno slice.