How to Make It
Preheat oven to 350°F. Generously coat a 9-inch springform pan with cooking spray. Place butter between two sheets of parchment paper. Using a rolling pin, beat butter to 1/4-inch thickness. Remove and discard top parchment paper sheet. Cut butter into approximately 1/3-inch pieces.
Place chestnut paste in bowl of a heavy-duty stand mixer fitted with paddle attachment. Beat on medium speed until smooth, about 1 minute. Add butter pieces; beat until smooth, about 1 minute, stopping to scrape down sides of bowl as needed. Add half of the beaten eggs; beat on medium speed until mixture separates and looks broken, about 20 seconds. Remove paddle attachment from stand mixer; attach whisk attachment. Add granulated sugar to mixture; whisk on medium speed until combined, about 30 seconds. Add egg yolks, rum, and remaining half of beaten eggs; whisk until smooth, about 30 seconds. Add almond flour and baking powder; whisk on low speed until just combined, about 30 seconds.
Pour half of the chestnut batter into prepared springform pan. Roughly chop 2/3 cup chestnuts; sprinkle over batter. Top with remaining batter, spreading evenly over chopped chestnuts. Cut remaining 1/3 cup chestnuts in half crosswise; gently press into top of batter.
Bake in preheated oven until cake is golden brown and center springs back when gently pressed, about 45 minutes. Cool cake in pan on wire rack 20 minutes. Remove sides of springform pan; let cake cool completely, about 2 hours. Sprinkle with powdered sugar. Slice cake, and dollop crème fraîche evenly onto servings.