Combine first 8 ingredients in a medium bowl. Fold in crabmeat. Shape crab mixture into 4 (4-inch) patties; chill 1 hour.
Melt butter with oil in a large skillet over medium-high heat; add crab cakes, and cook 4 to 6 minutes on each side or until golden.
Divide Radish-and-Avocado Salad among 4 serving plates; top each with 1 crab cake.