Position racks in the upper and lower thirds of the oven and preheat to 375. Line 2 baking sheets with parchment paper or silicone baking mats
In a heavy bottomed 2 quart pan, mix milk, 1/2 cup water, vanilla bean, butter, sugar and salt. Bring to a simmer over medium heat,s tirring occaswionally. Carefully remove vanilla bean and scrape any remaining seeds into liquid. Bring mixture to a boil. Add flour, lower heat to medium and immediately begin stirring vigorously with a wooden spoon. Don't let up til the dough starts to come away from the sides of the pan and form a loose ball. A light crust will form on the bottom of the pan. Keep stirring quickly to dry the dough, about 2 minutes more. The dough should now be very smooth and have lost most of its moist appearance
Transfer dough to bowl of stand mixer fitted with paddle attachment. Beat in eggs, 1 at a time, on medium speed. You'll think the dough is breaking apart and all is lost, but it will come together again, when all the eggs have been added. Beat in vanilla extract. Drop dough by the tablespoonful onto prepared baking sheets, leaving about 2 inches of space between dollops.
in a small bowl, beat together egg,vanilla extract, salt and 1 tsp. water til well blended. Brush each puff with this egg wash and sprinkle generously with pearl sugar
Bake for 15 minutes, then rotato the sheets from top to bottom and front to back. Continue baking til puffs are deeply golden and sound hollow when their crisp exteriors are tapped, another 15 to 20 minutes. Carefully transfer puffs to a wire rack to cool a bit before serving. Makes about 2 1/2 inch puffs