Using a food processor or a blender, process pecan halves until finely chopped.
1/2 cup pecan halves
In a large bowl, whisk together pecans, flour, brown sugar, baking powder, cardamom, and salt.
2 1/2 cups all-purpose flour, 1/2 cup brown sugar, 1 tablespoon baking powder, 1 teaspoon ground cardamom, 1/2 teaspoon fine sea salt
In another bowl, whisk together milk, egg, butter, and vanilla. Make a well in the center of the dry ingredients and pour milk mixture in. Stir to incorporate well. Cover and refrigerator for 1 hour.
1 cup milk, 1 large egg, 1/4 cup unsalted butter, 1 teaspoon pure vanilla extract
After dough has chilled, heat oil in a fryer or deep pan to 350°F-365°F.
Vegetable oil
Dust work surface with flour, and roll out dough to about 1/2-inch thick. Cut out donuts or donut holes using a round cutter, glass, or mouth of a bottle, dipping cutter into flour as needed. Gather scraps, re-roll, and cut until all remaining dough is used.
When fryer is hot, carefully drop donuts/donut holes into oil and let cook until deep golden brown, turning as needed (about 5 minutes total). Donut holes will turn on their own, so you might need to hold them in place with a fork to cook the less-browned side.
Remove from oil and let drain on paper bags or paper towels. Repeat until all donuts/donut holes are cooked.
Whisk together the powdered sugar, milk, and vanilla to make a glaze, adding more milk as needed to get desired consistency. Drop donut holes into glaze and lift out with a fork, tapping or shaking slightly to let excess glaze drip. Return donut hole to rack to let glaze set, and repeat with remaining donut holes.
3 cups powdered sugar, 4 to 5 tablespoons milk, 1 teaspoon pure vanilla extract
Serve immediately once glaze is set. Leftover donut holes will keep in a paper bag for 1-2 days.
Notes
Makes about 36 donut holes.
To make the original coffee variation, use cold coffee in the glaze instead of milk, and add 1/2 teaspoon espresso powder to the dough.