Cake: Preheat the oven to 350 degrees and position a rack in the center of the oven. Butter the bottom and sides of a 9-by-5-inch loaf pan. Place a long piece of parchment paper in the bottom of the pan, letting the parchment extend up the two long sides of the pan and overhang slightly. (This will make it easy to remove the pound cake from the pan after it is baked.) Butter the parchment paper and dust with flour; remove any excess flour.
Whisk together the flours, baking powder, and salt in a medium bowl.
Beat the butter and granulated sugar on high speed until fluffy, 4 to 5 minutes, in the bowl of a stand mixer fitted with the paddle attachment.
Add the vanilla and beat until incorporated, about 15 seconds. Scrape down the sides and bottom of the bowl.
With the mixer on medium-low speed, add the eggs and the egg yolks, one at a time, beating for about 1 minute after each addition. Scrape down the sides and bottom of the bowl again and beat for 10 seconds. Turn the mixer to low speed and add the flour mixture in three parts, alternating with the cream, beginning and ending with the flour mixture.
Transfer batter to prepared pan, smooth top with an offset spatula, and bake until a skewer inserted into the center comes out with a few moist crumbs, 55 to 65 minutes. (If top of cake begins to darken too much before it is done in the middle, tent pan with aluminum foil and continue baking.) Place pan on a cooling rack for 15 minutes. Using a small knife or offset spatula, gently loosen cake from sides of the pan, pull up on parchment paper to lift cake out of pan, and place it directly on cooling rack.
Salty Caramel Glaze: Stir together the brown sugar, cream, and butter in a medium saucepan over low heat. When butter is half melted, increase heat to medium-high and bring mixture to a boil. Whisking constantly, boil mixture for 1 1/2 minutes. Remove from heat, whisk vigorously for 1 minute to release excess heat, and add 3/4 teaspoon fleur de sel. Let cool 5 minutes.
Sift in 1/4 cup confectioners' sugar and whisk until combined. Continue adding confectioners' sugar, a tablespoon at a time, until the mixture is pourable and slightly thick (it will thicken more as it cools); you may not use all of the confectioners' sugar.
Place cake on wire rack set over rimmed baking sheet covered in parchment (to catch the excess caramel). Use a bamboo skewer to poke holes in cake. Pour warm caramel glaze over cake to cover. Sprinkle evenly with remaining 1 teaspoon fleur de sel. Let cool 15 to 20 minutes before serving.