Place the all-purpose flour in the bowl of a stand mixer fitted with the paddle attachment. Sift in the cake flour, baking powder, baking soda, and sugar and mix on the lowest setting for about 15 seconds to combine. Add the salt and mix to combine. Stop the mixer, add the butter, and on the lowest setting (to keep the flour from flying out of the bowl), pulse to begin incorporating th butter. Increase the speed to low and mix for about 3 minutes to break up the butter and incorporate it into the dry mixture. If any large pieces of butter remain, stop the mixer, break them up by hand, and mix until just incorporated.
With the mixer running, slowly pour in the cream. Add the crème fraîche and mix on low speed for about 30 seconds, until all of the dry ingredients are moistened and the dough comes together around the paddle. Scrape down the sides and bottom of the bowl and paddle, and pulse again to combine. Add the bacon, the 144 grams (2 cups) cheese, and the chives, and pulse to incorporate.
Mound the dough on a work surface (I used a silicone mat) and, using the heel of your hand or a pastry scraper, push it together. Place the dough between two pieces of plastic wrap (I used waxed paper instead) and, using your hands or a rolling pin, press it into a 7-by-9-inch block, smoothing the top. Press the sides of your hands against the sides of the dough to straighten them. Wrap the dough in plastic wrap (or bee's wrap) and refrigerate for about 2 hours, until firm.
Line a sheet pan with a Silpat or parchment paper. Cut the block of dough lengthwise in half, and then cut each half crosswise into 6 rectangles (70 grams each). Arrange them on the prepared sheet pan, leaving space between them. Cover with plastic (or bee's) wrap and freeze until frozen solid, at least 2 hours, but preferably overnight. (The scones can remain in the freezer for up to 1 month.)
Preheat the oven to 325°F (convection oven) or 350°F (standard). Line a sheet pan with a silpat or parchment paper (I just kept my scones on the same one they were on in the freezer.)
Arrange the frozen scones 1 inch apart on the sheet pan. Brush the tops with cream and sprinkle with remaining 36 grams (1/2 cup) cheese and black pepper. Bake for 24-27 minutes in a convection oven, 33-36 minutes in a standard oven, until golden brown. Set the sheet on a cooling rack and cool completely.
The scones are best the day they are baked, but they can be stored in a covered container for 1 day.
Source: Bouchon Bakery by Thomas Keller