In a small bowl, combine the warm water, honey and yeast. Allow to sit for 10 minutes to proof the yeast. It should be foamy.
In a large bowl, combine the flour, salt, olive oil and the yeast mixture. Stir with a wooden spoon until the dough begins to come together. Transfer to a lightly floured work surface and knead by hand until a soft smooth dough forms, about 5 to 8 minutes. Form the dough into a ball and place inside a lightly greased large bowl. Cover loosely with plastic wrap and a damp kitchen towel and allow to rest until doubled in size, for 2 hours.
Meanwhile, in a large bowl, cream together the cream cheese and 1/2 cup sugar on high for about 3 minutes. Stir in the vanilla and 1 tablespoon lemon juice until well combined. Chill in the fridge until ready to use.
In a separate bowl, toss together the sliced strawberries, raspberries, blueberries, remaining 1/4 cup sugar, and the remaining tablespoon of lemon juice. Allow to sit at room temperature to macerate until ready to use.
Once the dough has risen, gently punch it down and then transfer to a lightly floured work surface. Cut into three even pieces and form into balls. Keep them covered with a kitchen towel.
Working with one at a time, roll out into a rectangle or a circle. Brush the surface evenly with olive oil.
Preheat a large stovetop grill pan or an outdoor grill over medium heat. Once hot, carefully place the pizza dough onto the grill, oil side down. Then brush the top side with oil as well. Cook until the dough begins to puff up slightly and grill marks form, about 5 to 8 minutes. Carefully flip over and continue to grill on the second side for about 3 to 5 minutes. Transfer to a baking sheet and then continue grilling the rest of the pizza dough.
Spread the grilled pizza dough with the cream cheese mixture and then add the crumbled graham crackers. Spoon the berries over the top and garnish with fresh mint. Slice and serve immediately.