Crush the cookies or graham crackers into crumbs and stir through melted butter.
Beat the room-temperature cream cheese, vanilla, and condensed milk until well combined. If there are some lumps of cream cheese remaining, don't worry.
In a large bowl, whip the cream until stiff peaks form.
Fold the condensed milk mixture through the cream carefully until you have a thick and airy mixture.
Line a loaf tin with parchment paper, scoop ⅓ of the mixture into the lined loaf, and smooth it out.
Spoon on ⅓ of the lemon curd and use a skewer to swirl it into the ice cream. Crumble over ⅓ of the crushed cookies. Repeat this two more times with the remaining ingredients.
Freeze the finished ice cream for at least 6 hours.
To serve, let the ice cream sit at room temperature for 5 minutes to slightly soften before scooping.