In a large saucepan, combine half-and-half, sugar, reserved vanilla seeds, and vanilla extract. Cook over medium-high heat until mixture just begins to boil. Remove from heat and set aside.

In a large bowl, whisk egg yolks until smooth. Using a ladle, pour 1 cup hot vanilla mixture in a slow steady stream into yolks, whisking constantly. In the same manner, add 3 cups hot mixture, 1 cup at a time. Add egg mixture to saucepan, whisking well.

Cook over medium-high heat, stirring constantly, until thick enough to coat the back of a wooden spoon or until temperature reaches 170 degrees on an instant-read thermometer, being careful not to let it boil.

Pour custard into a fine-mesh sieve* and strain into a stainless-steel bowl set in an ice-water bath. Let cool to room temperature, stirring often. Cover custard well with plastic wrap or transfer to an airtight container. Refrigerate for 2 hours, stirring occasionally to prevent a skin from forming on custard.

Pour custard into the container of a 4-quart ice-cream freezer. Freeze according to manufacturer's instructions.

*If you do not have a fine-mesh sieve, cheesecloth works just fine.