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Place racks in the center and upper third of the oven and preheat to 350°F. Line two cupcake pans with papers.
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To make the cupcakes: In a large bowl, whisk together the flour, sugar, cocoa powder, baking soda, and salt.
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In a medium bowl, whisk together the oil, eggs, vanilla extract, vinegar, and cold coffee or water. Slowly whisk the wet ingredients into the dry, being careful not to over-mix. The mixture will be quite wet; that's OK.
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Pour the batter into the prepared pans until the cups are 2/3 full. Bake for 20 to 24 minutes, or until a toothpick inserted into the center comes out clean. Remove from the oven and cool in the pan for 10 minutes. Remove the cupcakes from the pan and place on a rack to cool completely before frosting
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To make the frosting: Place the butter and salt in the bowl of a stand mixer fitted with a paddle attachment. Beat on low until pale and very smooth, about 30 seconds.
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Add the confectioners' sugar 1 cup at a time, mixing until just incorporated. Stop the mixer and scrape down the sides of the bowl as necessary. Mix in the vanilla and milk and then beat the frosting on high speed until light and smooth, about 3 minutes.
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To assemble: Spread 1 1/2 tablespoons of frosting on each cupcake. Refrigerate until the topping is ready.
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To make the topping: In a medium saucepan over medium-low heat, combine the brown sugar, butter, cream, and salt. Bring to a simmer and cook until thick and bubbly, 3 to 5 minutes. Remove from the heat and stir in the vanilla and pecans. Rest for 5 minutes, then spoon over chilled, frosted cupcakes. The topping will harden as it cools.
