In another small pot, simmer remaining ¾ cup heavy cream, milk, sugar and salt until sugar completely dissolves, about 5 minutes. Remove pot from heat.
In a separate bowl, whisk egg yolks.
Whisking constantly, slowly whisk about a third of the hot cream into the yolks to temper the eggs, then whisk the yolk mixture back into the pot with the cream.
Return pot to medium-low heat and gently cook until mixture is thick enough to coat the back of a spoon (about 170º degrees F on an instant-read thermometer). Once ice cream base has come to temperature, remove from heat and stir in white chocolate mixture and peppermint extract.
Strain ice cream base through a fine mesh sieve into a clean bowl.
Cover with plastic wrap and chill at least 4 hours or overnight