Whisk together egg and milk.
Dip catfish fillets into egg mixture; then dredge in flour, shaking off excess.
Add olive oil to a heavy skillet, and heat to medium high. Add 2 fillets, rounded side down, and cook 3 to 4 minutes until golden brown. Turn fillets, and cook an additional 3 to 4 minutes or until done, and remove from skillet. Sauté remaining two fillets, and remove when done.
Chive Cream Sauce:
Add garlic and chives, and stir gently for 1 to 2 minutes. Deglaze skillet with vermouth. Slowly add cream, stirring constantly until sauce is reduced to desired consistency. Add salt and pepper to taste.
Serve sauce over catfish fillets.