For the cheesecake, mix the sliced strawberries, sugar, balsamic vinegar, and peppercorns together in a small saucepan. Bring to a boil over medium heat, then reduce heat to low and simmer for 25 to 30 minutes, until the strawberries have softened and released some of their juices. Remove from heat and allow the mixture to cool for 10 minutes before pureeing in a blender or food processor until smooth and the peppercorns are crushed. Set aside.
In the bowl of an electric mixer fitted with the paddle attachment, mix together the cream cheese, creme fraîche, and sugar at medium speed until smooth and fluffy, about 3 minutes. Reduce speed to medium low and add the eggs, one at a time, making sure each egg is completely incorporated and scraping down the sides of the bowl before adding the next egg. Add all but 1/3 cup of the strawberry puree to the cheesecake mixture and stir to incorporate. Set aside.
Preheat the oven to 325 degrees Fahrenheit and grease a 9-inch wide and at least 2 and 1/2-inch deep springform cake pan and line the bottom with parchment paper. Cut the parchment paper into strips as wide as the depth of your springform pan and line the sides with them, as well.
For the crust, mix together the graham cracker crumbs, sugar, and butter in a medium bowl until sand-like in texture. Empty the bowl into the bottom of the springform pan and press it into an even layer at the bottom. Place a sheet of tin foil on the bottom of a large casserole pan, then place the the springform pan onto the sheet and push the foil up around the bottom of it (this will help keep water from leaking into the springform pan). Pour water into the casserole pan so that it measures about 1-inch deep, taking care not to splash any into the springform pan. Pour the cheesecake mixture into the springform pan and place the whole casserole dish in the oven. Bake until the top of the cheesecake is golden brown and the cheesecake is still slightly jiggly but set, about 1 hour and 30 minutes.
Turn the oven off and leave the door ajar to allow it to gradually cool to room temperature, then remove the springform pan from the casserole pan and place the springform pan in the refrigerator for at least 4 hours. Remove it from the refrigerator and remove the sides of the springform pan and the side pieces of parchment paper. Spread the reserved 1/3 cup of the strawberry puree over the cheesecake and top with the desired amount of fresh strawberries. Serve immediately.