Position a rack in the center of the oven and heat the oven to 325 degrees F. Line the bottom and sides of a 9x13-inch straight-sided metal baking pan with heavy-duty aluminum foil, leaving about a 2-inch overhang on the short sides. Lightly butter the foil.
Put the butter in a large (4-quart) saucepan over medium-low heat and stir occasionally until melted, about 2 minutes. Off the heat, whisk in the cocoa powder until smooth, 1 minute. Add the sugar and salt, and whisk until well blended. Use your fingertip to check the temperature of the batter -- it should be warm, not hot. If it's hot, set the pan aside for a minute or two before continuing.
Whisk in the dark rum and vanilla extract and then the eggs, two and then three at a time, until just blended. Sprinkle the flour over the batter and stir with a rubber spatula until just blended. Stir in the pecans.
Scrape the batter into the prepared pan and spread evenly. Bake until a toothpick inserted in the center comes out with small bits of brownie sticking to it, 35 to 45 minutes. Do not overbake. Cool the brownies completely in the pan on a rack, about 3 hours.
When the brownies are cool, use the foil overhang to lift them from the pan. Invert onto a cutting board and carefully peel away the foil. Flip again and cut into 24 squares. Serve immediately or wrap in plastic and store at room temperature for up to 3 days. They can also be frozen in an airtight container or freezer bag for up to 1 month.
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