Bake the cheesecakes: Preheat oven to 325 degrees, and line the bottom and sides of 2 8x8-inch cake pans with parchment paper.
In a large bowl using a hand mixer on low speed, cream together the cream cheese, sugar and flour until smooth and combined. Add sour cream and vanilla and mix until combined. Add the eggs one at a time, scraping down the sides of the bowl after each addition. Mix in the mango puree.
Divide batter between the 2 cake pans. Place the pans inside another larger pan with high sides. Fill the outside of the pan with enough warm water to go halfway up the side of the cake pans. Bake 30 minutes, then turn the oven off and let the cheesecakes cool in the oven with the door slightly cracked, about 30 minutes. Remove cheesecakes from oven and chill until firm, at least 3 hours.
Assemble the sandwiches: Slice the cake loaf into 8 slices, and slice each chilled cheesecake into 4 squares.
Preheat skillet over medium heat. Butter one side of a pound cake slice, place cake butter-side down in pan, add 1 slice of cheesecake. Butter a second piece of cake and place on top of cheesecake in pan. Grill until lightly browned, then flip. Continue grilling until golden on the second side. Working in batches, repeat with the remaining 7 sandwiches. Serve with strawberry soup on the side.