In a small saucepan over medium heat, combine the milk and butter and heat, stirring, until the butter melts and the mixture is hot but not boiling (about 125°F/52°C on an instant-read thermometer). Remove from the heat.

Fit a stand mixer with the paddle attachment. In the mixing bowl, combine 2 1/2 cups (12 1/2 oz/390 g) of the flour, the brown sugar, cinnamon, salt, and yeast and beat on low speed to mix. Add the hot milk mixture, raise the speed to medium, and beat until well blended and smooth. Add the eggs and vanilla and beat until fully incorporated, about 2 minutes. Add the remaining 3/4 cup (4 oz/125 g) flour and beat until well blended and smooth, about 1 minute longer. The dough will not pull away from the sides of the bowl and will still be somewhat sticky, but your finger should come out clean when you insert it into the center.

Scrape the dough into a large bowl and cover the dough with a clean kitchen towel. Let stand until well risen and increased in bulk (it may almost double in size), about 45 minutes.

Line a baking sheet with waxed paper and brush the paper with oil. Line a second baking sheet with paper towels.

On a generously floured work surface, roll out the dough into a circle 10 inches (25 cm) in diameter and 1/2 inch (12 mm) thick. Using a 3-inch (7.5-cm) round donut cutter, cut out as many donuts as possible. Transfer the donuts and holes to the oiled paper. Gather up the donut scraps and repeat rolling and cutting. Cover the donuts and holes with a clean kitchen towel and let rise until soft and puffy, about 30 minutes.

Pour oil to a depth of 2 inches (5 cm) into a deep-fryer or deep, heavy saute pan and warm over medium-high heat until it reads 360 degrees F (182 degrees C) on a deep-frying thermometer. Carefully lower 2-5 donuts or holes into the hot oil and deep-fry until dark golden, about 1 1/2 minutes. Turn over and cook until dark golden on the second side, about 1 minute longer. Transfer to the towel-lined baking sheet. Repeat to fry the remaining donuts and holes, allowing the oil to return to 360 degrees F between batches.

Using a thin metal spatula, spread a thin layer of the caramel glaze over the top of each donut. (If the caramel is too thick, heat over low heat until spreadable, 2-3 minutes.) Spread the tops of the donut holes with a thin layer of glaze, or dip their tops in it. Dip the glazed portions of the donuts and holes in the toasted pecans. Press gently with your fingertips to help them adhere. Tap the donuts gently to shake off any loose pecan pieces. Sprinkle with the salt. Serve right away.

This donuts recipe is excerpted with permission from Donuts by Elinor Klivans and published by Weldon Owen.