Line a baking sheet with parchment paper. Use a 3-inch-round cookie cutter or glass to trace 12 circles on the parchment, leaving space between each. Flip the parchment over and lightly coat with cooking spray. Fit a pastry bag with a 1/2-inch round tip or snip a 1/2-inch hole in one corner of a large resealable plastic bag.
Sift the flour, baking powder, nutmeg and salt into a medium bowl. Beat the butter and sugar in a large bowl with a mixer on medium speed until well combined, about 2 minutes. Beat in the whole egg and yolk until incorporated and the mixture turns pale yellow, about 3 minutes. Beat in the milk and vanilla until smooth, scraping down the bowl as needed. Reduce the mixer speed to low; gradually beat in the flour mixture until just incorporated (do not overmix). Transfer the batter to the prepared pastry bag.
Using the circles as a guide, pipe the batter into 3-inch rings. (To make doughnut holes with the remaining batter, pipe 1-inch rounds onto the parchment.) Refrigerate while you heat the shortening, or cover with plastic wrap and refrigerate overnight.
Heat about 6 cups shortening in a large pot over medium-high heat until a deep-fry thermometer registers 340 degrees F. (You want to have about 2 inches of liquid once the shortening melts.) Use scissors to cut the parchment around each doughnut, leaving a 1-inch border. Carefully invert 2 doughnuts into the shortening and peel off the parchment with tongs. Fry until golden brown, 1 minute 30 seconds per side. (Fry the doughnut holes 30 seconds per side.) Remove with a slotted spoon and drain on paper towels. Bring the shortening back to 340 degrees F and repeat with the remaining doughnuts.