Preheat oven to 325 degrees F (160 degrees C). Grease and flour a 9-inch round cake pan and line the bottom of the pan with parchment paper.
In a small sauté pan, heat the Unsalted Butter (1 stick) over medium-low heat and cook until golden brown flecks form all around the edges of the pan. Watch the butter carefully because it will burn quickly! Remove the pan from the heat and using a rubber spatula, scrape the browned butter into a small bowl to cool, making sure to scrape up all the brown bits.
In a small bowl, combine the 3 tablespoons of Unsalted Butter , Maple Syrup (1 tablespoon), and Dark Brown Sugar (2 tablespoon) and stir to combine. Using a rubber spatula, spread the mixture evenly over the bottom of the pan.
Cut Bosc Pear (2) in half through the stem. Trim the stems and carefully core the pears with a melon baller or teaspoon. Cut two 1/2-inch thick slices (lengthwise) from each half so you end up with 4 slices from each pear. Artfully arrange the pears, cut sides down, on top of the butter mixture.
In a small bowl, combine the Hazelnuts (1 cup), All-Purpose Flour (1/2 cup) and Kosher Salt (3/4 teaspoon), and set aside.
In the bowl of a stand mixer (hand mixer works, too!), beat the Egg (6) and Granulated Sugar (1/2 cup) until the egg whites form firm peaks.
Using a rubber spatula, very gently fold in the browned butter, then the nut/flour mixture, until just combined. Transfer the batter to the pan, and smooth the top with a rubber spatula.
Bake for 35-40 minutes, until the cake is lightly browned around the edges and a toothpick inserted into the middle of the cake comes out clean.
Cool for 5 minutes and then carefully invert onto a serving plate.
To make unsweetened whipped cream, whip Heavy Cream (1 cup) with Vanilla Extract (1/2 teaspoon) in the bowl of an electric mixer fitted with a whisk attachment or in large mixing bowl using a hand mixer. Whip until just firm and serve immediately.
Serve warm or at room temperature with unsweetened whipped cream.