Preheat the oven to 350 degrees F. Arrange two racks evenly spaced in the oven.
Place the butter and brown sugar in the bowl of an electric mixer fitted with a paddle attachment and mix on medium speed just until the butter and sugar are combined so you don’t whip a lot of air into it. With the mixer on low, add the vanilla and scrape down the bowl with a rubber spatula. In a separate bowl, sift together the flour and salt. With the mixer on low, slowly add the flour mixture to the butter and sugar mixture and mix just until the dough comes together.
Scoop out 1-ounce pieces of dough (about a tablespoon), roll each one into a ball, then roll it into a 2 1/2-inch-long log. Place the logs 2 inches apart on two sheet pans lined with parchment paper and bake for 15 to 17 minutes, until the cookies start to brown around the edges and spring back when lightly touched. Set aside to cool on the pans.
For the coating, place the chocolate and the butter in a heatproof bowl and microwave on high for 30 seconds. Stir the mixture vigorously and continue to microwave in 30-second increments until the chocolate is just melted.
Drag half of the rounded top of each cookie through the chocolate and place it, chocolate side up, on a piece of parchment paper. While the chocolate is still warm, sprinkle on the pecans. Repeat with all the cookies. Set aside for at least 30 minutes for the chocolate to firm.