In a small pot, simmer cream, sugar and salt until sugar completely dissolves, about 5 minutes.
Remove pot from heat. In a separate bowl, whisk yolks.
Whisking constantly, slowly whisk about a third of the hot cream into the yolks, then whisk the yolk mixture back into the pot with the cream.
Return pot to medium-low heat, add vanilla, and gently cook until mixture is thick enough to coat the back of a spoon (about 170 degrees on an instant-read thermometer).
Add in cream cheese pieces and whisk until smooth.
Strain through a fine-mesh sieve into a bowl, using a spatula to press mixture through sieve.
Cool mixture to room temperature.
Cover and chill at least 4 hours or overnight.
Churn in an ice cream machine according to the manufacturer’s instructions.
Pour ½ ice cream into a loaf pan, add bourbon cherry reduction in dollops (I like to scoop them up with a fork, so most of the liquid remains and you just get those juicy, bourbon cherries), swirl with a butter knife; repeat with remaining ice cream and cherries.
Place in freezer for at least 4 hours to firm up.