Bring the water, to which you've added a tiny pinch of salt, to a simmer. Slowly pour in the semolina, whisking all the time in a single direction, as if you were making polenta. Continue simmering over low heat for a few minutes, stirring often to prevent scorching.
Transfer the cooked semolina to a large mixing bowl and let it cool. Mix in the sugar, egg yolks and whites, lemon zest and vanilla extract, then thin out the batter a bit with the milk.
Grease a 20mm (8 inch) springform pan with the butter. Then pour in the batter, flattening out the top a bit with a spatula. Bake in a pre-heated 190C/375F oven for 60 minutes or so, until golden brown on top.
Remove the migliaccio from the oven and let it cool completely. Serve topped with powdered sugar.