Whisk together brown sugar, cornstarch, and 1/4 tsp salt in a heavy medium saucepan, then whisk in milk and cream. Bring to a boil over medium heat, whisking frequently, then boil, whisking, 1 minute. Remove from heat and whisk in butter and vanilla. Pour into a bowl, then cover surface with buttered wax paper and chill until cold, at least 1 1/2 hours.