Bring a large pot of water to a boil, salt it generously, then add the sprouts. Cook them until just soft (not mushy), 5 to 10 minutes, depending on age and size. Drain sprouts and halve. Melt butter in a thick-bottomed pan over medium-low heat. Add shallots and cook until softened but not brown, about 5 minutes. Mix sprouts well with the shallots and the butter. Season to taste with salt and fresh ground white pepper.