Whisk milk, 1/4 cup sugar, the rose syrup, and salt in a medium saucepan. Bring to a simmer over medium heat. Remove from heat, and set aside.
Whisk yolks, cornstarch, and remaining 1/4 cup sugar in a bowl. Whisking constantly, gradually add milk mixture, 1/2 cup at a time. Pour back into pan. Cook, whisking, over medium-high heat until mixture thickens and registers 160 degrees on a candy thermometer, about 2 minutes.
Transfer to the bowl of an electric mixer fitted with the paddle attachment. Add butter; mix on medium speed until butter melts and mixture cools, about 5 minutes.
Cover with plastic, pressing it directly onto surface. Refrigerate at least 2 hours (up to 3 days). Let stand at room temperature 10 minutes and stir before using.