Step 1

Bring apple, orange zest, cinnamon stick, cloves, chancaca, figs, granulated sugar, aniseed, and 1¼ cups water to a boil in a medium saucepan over medium heat. Reduce heat to medium-low and simmer, stirring occasionally and scraping bottom of pan, until reduced by about a third, 40–50 minutes. Strain syrup into a medium bowl; let cool.Do ahead: Syrup can be made 1 week ahead. Cover and chill. Bring to room temperature before using.

Step 2

Bundle cardamom and aniseed in cheesecloth and tie with kitchen twine. Place sachet in a medium saucepan; pour in 2 cups water. Bring to a boil, then reduce heat to medium-low and simmer 5 minutes to infuse.

Step 3

Add sweet potatoes to pan; cook until fork-tender, 17–19 minutes. Remove sachet and drain sweet potatoes, reserving ¼ cup cooking liquid. Let cooking liquid cool slightly.

Step 4

Meanwhile, purée sweet potatoes in a food processor or blender until smooth. Scrape purée into a large bowl.

Step 5

Whisk yeast and sugar into reserved cooking liquid, then whisk into purée. Add flour and ¼ tsp. Diamond Crystal or Morton kosher salt. Mix with a wooden spoon until a sticky dough comes together, about 5 minutes. Cover bowl with plastic wrap and let dough rise in a warm, draft-free spot until doubled in volume, about 1 hour, or chill overnight.

Step 6

When ready to fry, fit a medium heavy pot with thermometer; pour in oil to come 2" up sides. Heat over medium-high until thermometer registers 325°.

Step 7

Stir remaining 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt and ¼ cup water in a small bowl until salt is dissolved. Dip your hands in salt water, then punch down dough. Pinch off a golf-ball-size piece, roll into a ball, and flatten slightly. Push your thumb through the center to create a hole. Carefully slide into oil. Stick chopstick or skewer through hole and spin doughnut around to widen hole slightly and set shape. Fry until deep orange-brown and crisp, about 5 minutes per side. Transfer to a wire rack set over a baking sheet. Repeat process with remaining dough, dipping your hands in salt water to prevent sticking and adjusting heat as needed to maintain oil temperature. (Pan should fit 3 or 4 doughnuts at a time.)

Step 8

Generously spoon syrup over doughnuts to coat.