Freeze the ice cream maker bowl for a full 24 hours before you start.
Warm the cream, milk, 1/2 cup of the sugar, and salt in a saucepan over medium heat until steam rises and small bubbles form at the edges. While it warms, whisk the egg yolks with the remaining 1/4 cup sugar until pale and thick.
Temper the yolks by pouring about 1/2 cup of the warm cream into the yolk mixture in a slow, thin stream, whisking constantly. Pour everything back into the saucepan.
Cook over medium-low heat, stirring constantly with a silicone spatula, until the custard coats the spatula and a finger dragged across it leaves a line that holds, about 170°F on an instant-read thermometer.
Strain the custard through a fine-mesh strainer into a clean bowl. Stir in the vanilla. Cool 30 minutes on the counter, then cover and refrigerate at least 4 hours or overnight.
Churn the cold base in your ice cream maker according to the machine instructions, usually 20 to 25 minutes, until thick like soft serve and doubled in volume.
During the final minute of churning, pour in the 1/3 cup Bailey's and let it blend for 30 seconds. This keeps the alcohol from interfering with the churn and preserves the whiskey flavor.
Pack the ice cream into a 2-quart container, press parchment paper directly onto the surface, cover, and freeze at least 2 hours to firm up.
Let the ice cream sit on the counter 5 to 10 minutes before scooping. Finish each bowl with a pinch of flaky sea salt.