In a medium mixing bowl, combine lemon juice, pickle brine, water, and sugar. Whisk until the sugar is fully dissolved.

Weigh the total liquid. Set aside ¼ of it. Calculate 0.2% of the total liquid weight and weigh out the corresponding amount of agar. Whisk the agar into the reserved portion of liquid until fully dispersed. Bring to a boil, then reduce to a rapid simmer for 5 minutes to hydrate and dissolve the agar completely.

Slowly add the rest of the pickle lemonade into the agar and pickle lemonade mixture. Whisk well to combine. Pour into a half hotel pan. Take care to make sure the pan stays level after putting it in the freezer. Freeze at least 6 hours, or until solid.

Loosen the frozen gel by forcibly pulling at the sides of the pan, rotating as needed. Invert onto a smooth dish towel or grade 100 cheesecloth. Rap the bottom of the pan to release the gel in one solid piece.

Transfer to a colander set over a mixing bowl. Let the pickle lemonade thaw through the cloth. Do not squeeze!

Pour Tajín onto a small plate and spread it out evenly. Moisten the rims of two tall glasses, then dip and gently rock each glass in the Tajín until coated. Fill the glasses to the brim with ice. Pour in the pickle lemonade, then garnish each glass with a skewered cornichon.