In a medium saucepan placed over medium heat, add the half-and-half and 1 1/2 cups of the heavy cream. Add sugar and salt and whisk until dissolved.
While the cream mixture heats, add 2 egg yolks to the remaining 1/2 cup of heavy cream, whisking to combine. When the cream is hot, slowly ladle some of the hot cream into the cream and egg yolks, whisking continuously. This tempers the eggs.
Then slowly pour the egg yolk mixture into the hot cream mixture, while whisking. Cook over medium-low heat until slightly thickened, being careful not to boil the cream mixture.
Strain the mixture into a heat-proof bowl. Whisk in the vanilla extract and vanilla bean paste. Cover with plastic wrap, pressing the plastic wrap directly on top of the cream mixture. Refrigerate until completely chilled.
Transfer to an ice cream machine and churn until the mixture is the consistency of soft serve. Transfer to a freezer-safe container. Freeze until firm, approx 3-4 hours.