Make the squash-infused bourbon: Place the oven rack in the center position and preheat the oven to 450°F. Place the whole butternut squash on a baking sheet and bake until blistered and soft enough that you can easily pierce the skin with a knife, about 1 1⁄2 hours. Remove from the oven and set aside until cool enough to handle but still quite warm.
Cut the squash in half lengthwise and scoop out the seeds and fibers from the center and discard them. Pull off the skin and discard it. Cut the flesh into 2-inch chunks. Place the pieces in a large, nonreactive container. Add the bourbon while the squash is still warm; this will help the infusion process. Cover and set aside in a dark area for 3 days, swirling the mixture occasionally. Taste the infusion; it should have a pronounced sweet squash flavor. If so, strain the liquid through a sieve, lined with a double layer of damp cheesecloth, discarding the solids. This infusion makes more than you need for just one cocktail, but it can be covered and refrigerated for up to 1 month.
Make the five-spice syrup: Add star anise, fennel seeds, cinnamon stick pieces, cloves, and peppercorns to a small, dry saucepan set over medium-high heat, shaking the pan often, until noticeably fragrant, about 2-3 minutes. Stir in sugar and water and heat until it just begins to reach a low boil. Adjust the heat to keep the liquid at a simmer for 5 more minutes, turn off heat and steep the syrup for an hour. Strain the syrup through a fine-mesh sieve then place it in a lidded container and store in the refrigerator. Bring to room temperature before using.
Make the cocktail: Add the prepared squash-infused bourbon, prepared five-spice syrup, and bitters to a cocktail shaker and fill the shaker with ice cubes; shake until well chilled. Double-strain through a fine-mesh sieve into an Old-Fashioned glass filled with three medium or one extra-large cube of ice. Twist, spritz, and garnish with a wide strip of orange peel.