Coat a rimmed baking sheet with cooking spray; set aside. In a medium nonstick saucepan, combine sugar, 1 cup water, corn syrup, cornstarch, glycerin, and salt. Bring to a boil over medium heat, stirring with a wooden spoon to dissolve sugar. Without stirring, cook until mixture registers 275 degrees (soft-crack stage) on a candy thermometer, about 8 minutes. Remove pan from heat; carefully stir in butter and extracts.
Pour hot mixture onto prepared baking sheet. Allow edges to cool and set slightly; using a metal spatula coated with cooking spray, fold edges toward the middle (this will help cool taffy). When taffy is cool enough to handle but still quite warm, stretch and pull with lightly buttered fingers, folding the taffy over onto itself before pulling again until just starting to firm up, about 20 minutes.
Before taffy cools and hardens too much, pull off ropelike pieces, and using kitchen scissors, make snips along sides. Gently stretch and pull snipped taffy on either end, forming thorny twigs. Snip into 7-inch lengths. Transfer to a parchment-lined baking sheet; let cool. Store the taffy twigs in an airtight container at room temperature up to 1 week.