STEP 1Line an 8 1/2- by 4 1/2-inch loaf pan with 2 pieces of parchment paper, leaving a 3-inch overhang on all 4 sides. Using an electric mixer, beat the condensed milk, cream cheese, and vanilla in a large bowl until combined. Add the heavy cream and beat on high speed until thick, stiff peaks form, 3 to 5 minutes.
STEP 2Meanwhile, using a food processor or blender, purée half the strawberries.
STEP 3Spoon half the vanilla cream mixture into the prepared pan, spreading it evenly. Tap the pan to get rid of air bubbles. Fold 1/3 of the strawberry purée into the remaining cream mixture until combined. Spoon remaining strawberry purée over the pink cream mixture and fold in, leaving streaks of the purée to create a swirled cream. Spoon over the vanilla cream in the pan. Gently tap the pan to distribute the mixture and get rid of any air bubbles (spreading will destroy the swirls). Cover and freeze for at least 8 hours and up to 2 weeks.
STEP 4Thirty minutes before serving, cut the remaining strawberries into pieces and toss with the sugar. Let sit, tossing occasionally, for 25 minutes. When ready to serve, invert the cake onto a platter and remove the pan and parchment. Spoon the berries (and any juices) over the top.