Preheat oven to 350 degrees F. Put all dry ingredients in a food processor, including walnuts and pulse until combined. Whisk together the egg yolk and water in a small bowl. Add the butter to the dry ingredients and cut the butter into the mixture by pulsing. Add the yolk and water and pulse until the dough just comes together. Remove dough to a flat surface and gently knead until it comes together into a cohesive mass. Wrap and chill in the refrigerator for at least 30 minutes. Roll the dough out on a lightly floured surface to 1/8-inch thickness. Cut the dough into 4-inch circles. Transfer the circles to a baking sheet and bake until lightly browned, about 12 to15 minutes. Let cool on a baking rack.
Melt the butter in a small saucepan on top of the grill. Add the sugar, cinnamon and salt and pepper and stir until the sugar is dissolved. Sprinkle fruit with salt and pepper. Grill the fruit on both sides, brushing with the butter mixture, until just cooked through. Remove the fruit to a platter.
Combine sugar and water. Cook over medium heat until sugar caramelizes to a dark golden brown. Slowly add the butter, then cream and whisk until smooth. Strain the sauce into a bowl and whisk in the rum. Keep warm.
Arrange grilled fruit on top of the cookies. Drizzle with the sauce and serve with vanilla ice cream or freshly whipped cream.