Melt butter in a large skillet over medium heat. Stir in onion and cook until softened and translucent, about 5 minutes. Add celery and bell pepper; continue cooking until vegetables are tender, about 5 minutes. Stir in mustard and cayenne pepper; cook for 2 minutes. Scrape vegetable mixture into a mixing bowl and refrigerate until cold.

Stir mayonnaise into chilled vegetable mixture along with soft bread crumbs, beaten egg, lemon zest, and parsley until combined. Gently fold in crab meat. Season with salt and pepper. Form crab mixture into 8 cakes, 1/2 inch thick.

Pour dry bread crumbs into a shallow bowl. Press crab cakes into bread crumbs one at a time until coated on all sides. Gently shake off excess crumbs and place crab cakes onto a plate. Cover with plastic wrap and refrigerate for 1 hour to overnight.

Make vinaigrette: Whisk together 3/4 cup vegetable oil, lemon juice, lemon zest, ginger, sugar, and salt in a small bowl until well combined. Store in the refrigerator until ready to use.

Toss mixed salad greens with vinaigrette in a mixing bowl until evenly coated and divide onto 8 serving plates. Arrange avocado slices over salads and set aside.