In a small saucepan, melt the butter. Remove the pan from the heat and stir in the maple syrup, brown sugar, spices, vanilla, and salt to combine.
Line a baking sheet with parchment paper or Silpat, and lay the puff pastry sheet on top.
Slice the persimmons in 1/4-inch thick slices along the equator. Coat one slice at a time with the maple syrup mixture, shaking off the excess liquid then layering the slices in a nice pattern on the puff pastry. Sprinkle the pecans over the top.
Bake for 15-20 minutes, turning the pan about 10 minutes into cooking. The dish is done when the persimmons soften and the puff pastry is lightly browned.
Remove from the oven and allow to cool at least 5 minutes. Garnish with pomegranate seeds and serve warm or at room temperature with an optional drizzle of crème fraîche.