Marinate tofu slices in pickle juice and 2 tablespoons of the oil for at least 4 hours or overnight.
When ready to fry, heat oil in a large (12-inch) skillet over low heat while preparing cutlets. Remove tofu from pickle juice, pat dry, and place on a small, rimmed baking sheet. Sprinkle both sides of each slab with spice rub and, working one at a time, dredge in flour, dip in thinned Dijon, and dredge in panko, shaking the flour and panko pans as you dredge to help coat them.
Increase oil to medium-high. When a pinch of panko dropped into the hot oil sizzles immediately, you are ready to fry. Add cutlets to hot oil and cook, turning only once, until golden brown on each side, about 6 minutes total.
Spread each bun, toasted or not, with mayonnaise and top with lettuce, cutlet, pickle slices, and a few shakes of hot sauce, if desired. Serve.