Make the Beurre Noisette:
Place butter in a light-colored saucepan so you can see the color change. Cook over medium heat, swirling occasionally.
Butter will melt, foam vigorously, then subside. Watch the milk solids at the bottom—they'll turn from white to golden to amber brown.
When butter smells nutty and toasty and the solids are deep amber (not black), immediately remove from heat and pour into a heatproof bowl to stop cooking. This takes 5-8 minutes total.
You need about 1¼ cups total: ¾ cup for the cake, ½ cup for the buttercream. Cool to room temperature before using.

Make the Spiced Pear Compote:
Combine diced pears, brown sugar, lemon juice, ginger, cinnamon, cardamom, cloves, pepper, and salt in a medium saucepan.
Cook over medium heat, stirring occasionally, until pears soften and release their juices, then continue cooking until mixture is thick and jammy, 15-20 minutes. Some pear pieces should break down while others hold their shape.
Remove from heat and stir in pear brandy or vanilla. Cool completely. Compote will thicken further as it cools.

Make the Spiced Ginger-Pear Sponge:
Preheat oven to 350°F (175°C). Grease two 9-inch round cake pans and line bottoms with parchment.
Whisk together flour, baking powder, baking soda, ground ginger, cinnamon, cardamom, nutmeg, cloves, and salt in a large bowl.
In a stand mixer with paddle attachment, beat room-temperature beurre noisette and brown sugar until well combined and slightly fluffy, 2-3 minutes.
Add eggs one at a time, beating well after each. Add vanilla and fresh grated ginger and mix until combined.
Add flour mixture in three additions, alternating with buttermilk (beginning and ending with flour), mixing on low just until combined after each addition.
Fold in diced pears by hand, distributing evenly throughout the batter.
Divide batter evenly between prepared pans and smooth tops.
Bake 30-35 minutes until golden, springy to touch, and a toothpick inserted in center comes out clean (a few moist crumbs are fine; wet batter is not).
Cool in pans 15 minutes, then turn out onto wire racks to cool completely.

Make the Beurre Noisette Buttercream:
Combine egg whites, sugar, and salt in the bowl of a stand mixer. Set over a pan of simmering water (bowl should not touch water).
Whisk constantly until sugar dissolves and mixture reaches 160°F (71°C)—it should feel completely smooth when rubbed between fingers.
Transfer bowl to stand mixer with whisk attachment. Beat on high until stiff, glossy peaks form and bowl is cool to touch, 10-12 minutes.
Switch to paddle attachment. With mixer on medium-low, add softened butter one tablespoon at a time, beating well after each addition.
With mixer running, slowly drizzle in the cooled beurre noisette. The buttercream may look curdled briefly—keep beating until it comes together smooth and silky.
Add vanilla and cinnamon. Beat until fully incorporated. If too soft, refrigerate 15 minutes and beat again.

Assemble the Cake:
Level cake layers if domed, using a serrated knife.
Place first layer on cake plate or board. Spread a generous layer of beurre noisette buttercream, leaving a ½-inch border.
Spoon half the pear compote over the buttercream, spreading gently. The buttercream border acts as a dam.
Top with second cake layer, bottom-side up for a flat top. Press gently to adhere.
Apply a thin crumb coat of buttercream over the entire cake. Refrigerate 30 minutes until firm.
Apply remaining buttercream in a smooth, even layer, or create rustic swoops and texture with an offset spatula.

Finish the Cake:
Toss thin pear slices with lemon juice to prevent browning.
Arrange pear slices in an overlapping fan or cascade on top of the cake.
Drizzle with honey.
Press crushed gingersnaps around the base of the cake.
Finish with flaky sea salt and edible gold dust if using.