Stir together half-and-half, brown sugar, and cornstarch in a 2- to 3-quart heavy saucepan and heat over moderate heat, stirring occasionally, until sugar is dissolved and mixture is heated through (do not let boil).
Whisk together yolks and salt in a medium bowl until smooth, then add hot half-and-half mixture in a slow stream, whisking. Return mixture to saucepan and cook over moderate heat, whisking, until mixture just reaches a boil, then boil, whisking, 1 minute (mixture will thicken). Immediately pour through a fine-mesh sieve into a clean bowl and stir in butter and vanilla until butter is incorporated. Set bowl in a larger bowl of ice and cold water and chill, stirring frequently, until pudding is cool, about 10 minutes.
Beat heavy cream with sour cream in a bowl with a whisk until mixture just holds stiff peaks.
Divide pudding among 6 small bowls, then top each serving with a generous dollop of cream.
