For the doughnuts: Line a rimmed baking sheet with parchment. Use a 2 1/2-inch round cookie cutter or glass to trace 16 circles on the parchment, leaving equal space between each. Flip the parchment over and lightly coat with cooking spray. Fit a large 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.
Whisk together the flour, baking powder and salt in a medium bowl. Set aside.
Pulverize the sugar and freeze-dried strawberries in a blender or food processor until no large pieces remain, then transfer to a large bowl. Add the butter and beat until combined (it will be crumbly). Beat in the whole egg and yolk until well combined and the mixture turns pale yellow, 4 to 5 minutes, scraping down the bowl as needed. Beat in the buttermilk, vanilla extract and strawberry extract until smooth.
Add half of the flour mixture to the wet ingredients and fold in with a rubber spatula until almost combined. Add the remaining flour mixture and fold in until just combined (do not overmix). Transfer the batter to the prepared pastry bag.
Using the circles as a guide, pipe the batter into 2 1/2-inch rings. (If you have any leftover batter, pipe 1-inch round doughnut holes on a second sheet of parchment sprayed with nonstick spray.) Refrigerate while you heat up the shortening or oil.
Heat the shortening or oil in a large pot over medium-high heat until a deep-fry thermometer registers 340 degrees F. (You want to have at least 1 inch of liquid once the shortening melts; about 6 cups.) Set a wire rack in a rimmed baking sheet.
Cut the parchment around each doughnut with kitchen shears, leaving a narrow border if possible. Carefully invert 4 of the doughnuts into the oil and peel off the parchment with tongs. Fry until golden brown and cooked through, about 2 minutes total, flipping the doughnuts every 30 seconds to ensure even cooking. (Fry any doughnut holes for about 1 minute.) Remove with a slotted spoon and drain on the wire rack. Bring the shortening back to 340 degrees F and repeat with the remaining doughnuts. Let the doughnuts cool slightly.
Meanwhile, make the glaze: Combine the confectioners' sugar, strawberry preserves, lime zest and lime juice in a medium bowl. Whisk until smooth and the consistency is like thick glue.
While the doughnuts are still warm, dip one at a time into the glaze, swirling as needed to coat the top. Return to the wire rack. Toss any doughnut holes in the glaze to coat completely and return to the wire rack. Let the glaze set, about 15 minutes.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)