In a shallow bowl or pie plate, whisk together egg and milk. Spread flour and remaining 2 cups panko in 2 separate shallow bowls or pie plates.
Form patties using about 1/4 cup crab mixture for each; cover evenly in flour. Dip patties in egg mixture, then coat both sides with panko; transfer to large platter. Repeat process to make 12 to 14 crab cakes. Refrigerate 30 minutes.
Preheat oven to 400°F. In a large nonstick skillet, heat 1 tablespoon oil over medium-high heat. Add 4 or 5 crab cakes and cook 2 minutes per side or until golden. Transfer to a baking sheet. Repeat twice with remaining crab cakes, using 1 tablespoon oil per batch. Bake until heated through, 8 to 10 minutes.
In a large saucepan, add beets and enough water to cover; bring to a boil over high heat. Reduce heat, cover, and simmer until tender, about 15 minutes (30 minutes for medium beets). Drain; rinse with cold water. Once cool, peel each beet and cut into quarters (cut medium beets into 1-inch chunks). In a small bowl, whisk together lemon juice, 2 teaspoons olive oil, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Add beets and toss; set aside.
In a medium saucepan over medium heat, simmer wine and vinegar until mixture reduces to a light syrup, about 5 minutes. Reduce heat to low and, whisking constantly, add butter, one piece at a time until well incorporated (be careful not to allow sauce to boil). Remove pan from heat; whisk in lemon juice, salt, and pepper.
Divide butter sauce in half. To make grainy mustard sauce: Whisk in mustard. To make herbed lemon butter sauce: Whisk in chives. Makes 1/3 cup of each.
Transfer crab cakes to warm serving plates. Garnish with baby beets and lemon wedges. Spoon mustard sauces around crab cakes and beets; serve immediately.