Line a 9-inch springform pan with parchment paper.

In a medium saucepan, melt butter over medium heat. Cook until butter turns a medium-brown color and has a nutty aroma, about 10 minutes. Pour into a heatproof bowl; in a small microwave-safe bowl, reserve 1 tablespoon (12 grams) browned butter for glaze.

In the bowl of a stand mixer, whisk together 1½ cups (188 grams) flour, 2 tablespoons (24 grams) granulated sugar, yeast, and 1½ teaspoons (4.5 grams) salt.

In a medium saucepan, heat ¼ cup (50 grams) browned butter, ⅓ cup (80 grams) milk, and ⅓ cup (80 grams) water over medium heat until an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add hot butter mixture to flour mixture; using the paddle attachment, beat at medium speed until combined. Add egg, beating until combined. With mixer on low speed, gradually add remaining 1¼ cups (156 grams) flour, beating just until a shaggy dough forms.

Switch to the dough hook attachment. Beat at low speed until dough is soft and elastic, 6 to 7 minutes, stopping to scrape sides of bowl and dough hook. Cover and let stand in a warm, draft-free place (75°F/24°C) for 10 minutes. (Dough will puff slightly.)

In a medium bowl, stir together pecans, remaining ⅓ cup (67 grams) granulated sugar, and remaining ¾ teaspoon (2.25 grams) salt.

Turn out dough onto a lightly floured surface. (Do not punch down.) Divide dough into 14 portions (about 45 grams each) and press each portion into a roughly round shape. (It does not need to be perfect.) Dip each in remaining browned butter, and roll in pecan mixture. Arrange evenly in prepared pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 30 to 40 minutes.

Preheat oven to 350°F (180°C).

Evenly sprinkle any remaining pecan mixture onto dough.

Bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), 25 to 30 minutes. Let cool in pan for 10 minutes. Remove from pan, discarding parchment.

Microwave reserved 1 tablespoon (12 grams) browned butter on high in 10-second intervals until melted. Add confectioners’ sugar, vanilla, and remaining 1 to 2 tablespoons (15 to 30 grams) milk, whisking until combined and desired consistency is reached. Spoon glaze into a pastry bag, and cut a ⅛-inch opening in tip. Drizzle glaze onto warm rolls. Serve warm or at room temperature.