Pulse the bread in a food processor until finely ground. Scoop 1 cup of the bread crumbs into a medium bowl and spread the remaining bread crumbs (about 3/4 cup) on a large plate. Grate the zest of the lemon into the bowl along with the breadcrumbs.
Pulse the fish in the food processor until coarsely diced (do not form a paste). Add the fish to the bowl with the lemon-bread crumbs along with the pickles, mayonnaise, egg, 3/4 teaspoon salt, and a few grinds of pepper. Form into 4 patties and dredge in the bread crumbs on the plate, turning to coat both sides.
Heat 3 tablespoons of the oil in a large heavy skillet over medium heat. When hot, add the fish cakes and cook, flipping once, until golden and cooked through, 8 to 10 minutes.
Halve the lemon and squeeze half of the juice into a bowl. Cut the other half into wedges. Toss the baby greens with the lemon juice and remaining 2 tablespoons oil. Season with 1/2 teaspoon salt and a few grinds of pepper. Serve the fish cakes with the salad and lemon wedges.
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