Heat milk until steaming and steep jasmine tea for 5 minutes, then remove. Whisk egg yolks, sugar, cornstarch, and salt until smooth. Slowly whisk in the warm milk. Return to the saucepan and cook over medium-low heat, whisking constantly, until thick like pudding and just starting to bubble. Cook 30 seconds more, then remove from heat. Let sit 1 minute, then stir in butter and vanilla. Strain if needed, cover the surface, and chill completely.