In a large bowl using a hand blender, or in bowl of stand mixer, beat cream cheese and sugar until fluffy. Add in sweetened condensed milk and vanilla bean paste, and mix to combine.
In separate bowl, whip heavy cream to stiff peaks. Fold into cream cheese mixture 1/3 at a time.
Add 2/3 of the graham cracker crust pieces and fold into ice cream. Pour half of ice cream into standard loaf pan. Sprinkle half of remaining crust onto ice cream. Add remaining ice cream, then remaining crust pieces. Freeze for 24 hours (cover with plastic wrap or parchment after 4 hours).