Preheat the oven to 250 degrees F.
For the lamb: Cut the meat with the grain into 1/2-thick slices. Season each piece on both sides with the salt and pepper. Place the flour in a pie pan and mix in the garlic powder and cayenne pepper. Place the eggs in a separate pie pan and beat. Dredge the meat on both sides in the egg, and then the flour. Repeat with all the meat pieces. (Reserve the remaining flour for the pan gravy.) Place the meat on a plate and allow it to sit for 10 to 15 minutes before cooking.
Place enough of the vegetable oil to cover the bottom of a 12-inch skillet and set over medium-high heat. Once the oil begins to shimmer, add the meat in batches, being careful not to overcrowd the pan. Cook each piece on both sides until golden brown, approximately 4 minutes on each side. Remove the lamb to a wire rack set in a half sheet pan and place in the oven to stay warm while the other pieces are being prepared. Repeat until all the meat is browned.
For the gravy: Add 1 tablespoon vegetable oil to the pan. Add the white wine to deglaze the pan and reduce by half. Whisk in 2 tablespoons of the reserved flour. Then whisk in the Dijon mustard and the vegetable stock. Continue to whisk until the gravy comes to a boil and begins to thicken. Add the milk and thyme and whisk until the gravy coats the back of a wooden spoon, 5 to 10 minutes. Season to taste with more salt and pepper, if needed. Serve over the lamb steaks.