Combine the roasted chestnuts, milk, cinnamon stick, and salt in a small saucepan. Simmer on low for about 20 minutes until the milk is infused. Strain the chestnuts, reserving the milk. Puree the chestnuts in a food processor, adding a tablespoon of the milk mixture at a time to help create a smooth puree (I used about 5 tablespoons). Transfer the leftover milk to a liquid measuring cup and add enough heavy cream to equal 2 cups total. Transfer this liquid to a small saucepan.
Over medium-low heat, bring the liquid up to just barely a simmer (be careful not to boil). Meanwhile, vigorously whisk the egg yolks in a large bowl until pale in color. Whisk in the brown sugar, granulated sugar, and ¼ cup of the chestnut puree. When the liquid has heated up, pour half of it into the egg mixture in a slow, steady stream. This will temper the eggs. Then, pour the tempered egg mixture back into the saucepan. Cook on medium-low heat until the custard thickens and coats the back of a spoon. Pour the custard into a bowl and stir in the vanilla extract. Let cool slightly before serving.
Whisk the flour, brown sugar, baking powder, salt, and nutmeg together in a large bowl until incorporated. In a separate bowl, beat the eggs. Add the milk, vanilla, and melted butter to the eggs. Whisk to combine. Pour the wet ingredients into the flour mixture, whisking until the flour soaks up the liquid and the mixture is homogenous. Heat a griddle or skillet over medium heat. Working in batches, add a tablespoon of butter at a time to the griddle or skillet and let melt (it will take only seconds). Ladle the pancake batter onto the surface, creating your desired size pancakes and evenly spacing them apart. Cook on one side until golden brown and bubbles begin to form on the top. Using a spatula, flip the pancake over and finish cooking on the other side. Keep the pancakes warm as you finish cooking all of the batter.
Spiced Pear Syrup
Meanwhile, heat a large sauté pan over medium-high heat. Add the butter. Just when then butter begins to brown, throw in the sliced pears. Sauté for a few minutes, until the pears begin to soften. Decrease the heat to medium-low and stir in the brown sugar. Add the water, cloves, cinnamon stick, star anise, and salt. Stir to combine and let the syrup simmer until smooth. Serve warm.
Spread the chestnut cream onto a plate. Stack the pancakes. Spoon out some of the pears and drizzle the syrup all over the top.
If you are roasting the chestnuts yourself, first preheat the oven to 425°F. Using a very sharp knife, cut an X in each chestnut, all the way through the skin and just piercing the flesh. Roast the chestnuts until golden and the skins start break away from the flesh, about 10-15 minutes. Keep a close eye on them as they roast. Depending on your oven and the chestnuts, they will take different times to cook.
You will have extra chestnut puree. Refrigerate in a sealed container for making these pancakes again or for another use (you could add it to soups or other purees).