Pour the milk into a large, heavy-bottomed saucepan and warm it over medium heat, being careful not to let it boil.
Slowly sprinkle the semolina into the warm milk while continuously stirring. Add the sugar and keep stirring so the mixture doesn't lump.
Turn the heat down to low and continue stirring as the semolina thickens, about 10-15 minutes, until it reaches a creamy consistency. Remove from the heat and stir in the vanilla extract.
Pour into serving bowls and allow to cool slightly. Serve warm or cold, traditionally with a dollop of jam, or with fresh berries or a sprinkle of cinnamon.