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Cook butter in a small heavy saucepan over medium heat, stirring occasionally, until it has a nutty fragrance and flecks on bottom of pan turn golden, 3 to 4 minutes. Transfer butter to a bowl and chill until just firm, about 1 hour.
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Beat together butter and brown sugar with an electric mixer until pale and fluffy. Beat in vanilla, then mix in flour and salt at low speed until just combined. Transfer dough to a sheet of wax paper or parchment and form into a 12-inch log, 1 1/2 inches in diameter. Chill, wrapped in wax paper, until firm, about 1 hour.
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Preheat oven to 350°F with rack in middle. Slice dough into 1/4-inch-thick rounds, arranging 1 1/2 inches apart on an ungreased baking sheet. Bake until surface is dry and edges are slightly darker, 10 to 12 minutes. Transfer to a rack to cool.