Put the skate fillets in a dish large enough to hold them in one layer. Pour the milk over them and sprinkle with salt and pepper to taste. Turn the fillets in the milk so they are coated on both sides.
Shake excess milk off the fillets. Lightly dredge the fillets in the flour on both sides.
Heat the oil in a large nonstick skillet over medium-high heat. When the oil is hot, add the fillets. Saute on one side until golden brown, about 3 minutes. Turn and cook on the other side until golden brown, about 3 minutes more. When the fillets are done, transfer them to a warm platter or serving plate.
Wipe out the pan with a paper towel and return the skillet to the heat. Melt the butter and add the shallots and garlic. Cook until shallots are softened. Add the sun dried tomatoes and cook about 2 minutes or until softened. Add chicken broth, tarragon, parsley, and mustard. You can thicken it a little with some flour if you desire. Cook briefly; then stir and pour the sauce over each fillet evenly. Serve immediately.