Pour the milk into a 2-quart heavy-bottomed saucepan and set over medium heat. Warm until steaming and small bubbles appear at the edges, about 4 to 5 minutes. Do not let it boil.

Stir in the rolled oats and brown sugar. Reduce heat to medium-low and cook, stirring often, for 12 to 15 minutes until the oats are soft and the mixture has thickened noticeably.

While the oats cook, whisk the egg yolks and salt together in a medium bowl until pale and slightly thickened, about 1 minute.

Temper the yolks: ladle about 80 ml of the hot oat mixture into the yolks while whisking constantly. Then ladle in a second scoop and whisk again. This prevents scrambling.

Pour the tempered yolk mixture back into the saucepan in a slow stream, stirring the pot constantly. Cook over medium-low heat for 3 to 4 minutes, stirring without stopping, until the pudding coats the back of a spoon and leaves a clean line when you drag your finger across it.

Remove from heat and stir in the vanilla extract and butter, if using. Stir for 30 seconds until the butter melts and the pudding looks glossy.

Spoon into bowls to serve warm, or pour into small glasses, press plastic wrap directly onto the surface, and refrigerate for at least 2 hours to serve chilled.