Preheat oven to 400 °F. Melt the butter in a frying pan over high heat. Seasonpigeon halves with salt and pepper. Brown one side. Turn the halves over and place the pan in the oven. Bake for 7 minutes.
Place the baked pigeon halves on a plate, tent with foil, and set aside to rest for five minutes. While the pigeon is resting, continue to reduce the sauce until thickened. Taste the sauce and season with salt, as required.
When the meat has rested, slice each half on the bias in four places. Arrange the pigeon slices on the serving plates. Top with the reduced sauce. Serve with vegetables.