In a large saute pan, over medium heat, add the oil. When the oil is hot, saute the skate for 2 to 3 minutes on each side. Remove the fish from the pan and set aside.
Add the shallots and capers. Season with pepper. Saute for 1 minute. Add the lemon juice and wine. Bring the liquid to a boil. Reduce the heat to medium low and simmer until the liquid reduces by half, about 6 to 8 minutes. Whisk in the butter, a cube at a time. Season with salt and pepper. Stir in 1 tablespoon of parsley.
Add the fish back into the sauce and simmer for 2 to 3 minutes.
In another saute pan, heat the remaining tablespoon of the oil. When the oil is hot, add the vegetables. Season with salt and pepper. Saute for 2 to 3 minutes.