Step 1 Melt one stick of butter in a heavy-bottomed saucepan until the milk solids have turned a rich brown color and the butter smells nutty. Watch carefully so it doesn't burn. Cool in a small bowl and then chill for a couple of hours in the fridge. Leave the second stick of butter out at room temperature.
Step 2 Heat the oven to 350 degrees. Spread the pecans on a rimmed baking sheet and toast for a few minutes until they are fragrant and darker in color. Cool, then finely grind using a food processor (be careful not to overwork the pecans, or they will turn to a butter).
Step 3 Sift together the flour and salt in a separate bowl.
Step 4 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl using a hand mixer, cream the browned and regular butters together with the brown sugar until light and fluffy, 3 to 5 minutes. Add the vanilla and pecans and beat until well combined. Add half the flour until just combined, and then the rest of the flour. Don't overbeat at this point. The dough will be kind of shaggy.
Step 5Dump the dough out onto a sheet of plastic wrap or wax paper and gather into a large ball. Cut the ball in half and use wax paper or plastic wrap to form logs 2 to 3 inches in diameter. Mix together turbinado sugar and fleur de sel, and roll the logs in the mixture, pressing it into the dough. Wrap the logs in plastic wrap or wax paper and refrigerate until thoroughly chilled, at least 2 hours and preferably overnight.
Step 6 Heat the oven to 350 degrees. Slice the logs crosswise into one-fourth-inch rounds. Space the cookies approximately 2 inches apart on a parchment- or silicone baking mat-lined baking sheet. Bake until set and lightly colored around the edges, about 12 minutes, rotating the pan halfway through.
Step 7The cookies will be very delicate just out of the oven. Cool them on the baking sheet before removing.