Melt butter in a saucepan over medium heat, swirling until deep golden and nutty, about 3 to 4 minutes. Stir in sugar and salt; cook 2 minutes until it resembles wet sand. Slowly add milk, whisk smooth, then add cream and corn syrup. Bring to a bare simmer.

Whisk yolks in a heatproof bowl. Ladle in about 1 cup of the hot mixture while whisking, then return everything to the saucepan.

Cook over medium-low, stirring constantly, until the custard coats the back of a spoon (170 to 175°F), about 6 to 8 minutes. Do not boil.

Remove from heat, stir in vanilla, and strain into a clean bowl. Press plastic wrap on the surface. Cool at room temperature 30 minutes, then refrigerate until very cold, at least 4 hours.

Churn in an ice cream maker until the texture of soft-serve, 20 to 25 minutes.

Transfer to a container, press parchment on the surface, seal, and freeze until firm, at least 4 hours. Scoop and serve.