In a large glass pitcher, pour the Harry & David Bloody Mary mix, hot sauce, cornichon juice, and Dijon mustard. Stir thoroughly, but don't shake. Add the vodka. Refrigerate until cold, about 1 hour.
Scatter the chili-lime seasoning onto a plate. Working with one glass at a time, run a lemon wedge along the rim of the glass. Turn the glass upside down onto the plate to coat the rim with the seasoning, shaking off any excess. Repeat with the remaining glasses, saving the lemon wedges as a garnish, for each glass, if you like.
Working with one glass at a time, fill them with ice. Place a celery stalk in the center of the glass, then fill with the chilled bloody mary mix.
Skewer the green olives and cornichons and serve with each glass.