1. Make dough: Stir yeast into 3 tbsp. warm (110°) water in bowl of a stand mixer or a large mixing bowl. Let stand until softened, about 5 minutes. Add milk, granulated sugar, salt, butter, egg, cardamom, and 2 cups flour; mix with dough hook on low speed or stir until combined. Blend in about 1 more cup flour (enough for a soft dough).
2. Mix with dough hook on medium speed until dough is smooth, elastic, and no longer sticky, 8 to 10 minutes, adding more flour if needed so dough pulls cleanly from bowl. (Or turn out onto a lightly floured work surface and knead, adding flour as required to keep dough from sticking.)
3. Oil bowl, add dough and turn over, then let rise, covered, in a warm place until doubled, about 1 hour. Punch down dough and divide into 2 equal parts.
4. Make filling: In a small bowl, combine brown sugar, cinnamon, cardamom, apricots, almonds, and chocolate.
5. Roll each dough portion on a lightly floured work surface into a 7- by 14-in. rectangle with a long side facing you. Spread rectangles with butter and sprinkle with apricot mixture to within 1 in. of the edge farthest from you. Starting at nearest side, roll each rectangle into a log and seal along edge (but not ends) with a good pinch. Line a large baking sheet with parchment paper and put logs next to each other on paper, seam down.
6. Weave logs together to form a loose braid. Lift up braid and gently twist once or twice. Form braid into a ring on baking sheet, pinch ends together to join, and tuck under. Cover loosely with a kitchen towel and let rise in a warm place until puffy, 30 to 45 minutes.
7. Preheat oven to 350°. Add toppings: Brush ring with egg and scatter with almonds. Bake until golden brown and an instant-read thermometer inserted in bread reads 185° to 190°, about 30 minutes. Slide cake from parchment to a rack. Let cool at least 20 minutes.
8. Whisk powdered sugar and syrup in a bowl until smooth. Drizzle over cake.
Make ahead: Up to 1 day, chilled. Set on a baking sheet, wrap with foil, and rewarm in a 250° oven 10 to 15 minutes.
Note: Nutritional analysis is per serving.