1. To make the candied beets: Divide simple syrup evenly between three medium saucepans. Bring to a boil over medium heat. Add each variety of beets to a different saucepan and bring to a simmer. Reduce heat to low and simmer until beets are translucent, about 1 hour. Remove from heat and let cool completely.

2. Line a baking sheet with a nonstick baking mat or parchment paper. Place sugar in a shallow dish. Gently drain beets, taking care not to break them, and dredge in sugar. Transfer to baking sheet and place in a cold oven to dry overnight. Beets may be kept in an airtight container for up to 1 week.

3. To make the cake: Preheat oven to 425 degrees. Trim the tops from beets, leaving about 1 inch of the stems. Line a large piece of aluminum foil with a piece of parchment paper; make a bed of coarse salt in the center. Place beet on salt and tightly wrap foil to enclose. Transfer to oven and roast until beets are very tender, about 1 hour.

4. When beets are cool enough to handle, peel and cut into large pieces. Transfer to the bowl of a food processor; process until very smooth. Strain and measure out 1/2 cup for the cake; reserve remaining beet puree for the sauce.

5. Reduce oven temperature to 350 degrees. Butter a 9-by-9-inch square baking pan; coat with sugar, tapping out excess, and set aside.

6. Sift together flour, baking soda, and 1/8 teaspoon salt onto a piece of parchment paper; set aside. In the bowl of an electric mixer fitted with the whisk attachment, beat together sugar, egg, and oil on medium-high speed until pale and light. Beat in vanilla and slowly add flour mixture; beat until well combined. Add chocolate and beat until well combined. Add 1/2 cup beet puree and beat until well combined.

7. Pour batter into prepared pan and rap on the work surface to remove air bubbles. Transfer to oven and bake, rotating pan after 10 minutes of baking, until a cake tester inserted into the center comes out clean, 18 to 20 minutes total. Let cool completely before removing from pan.

8. To make the raspberry-beet sauce: Place the raspberries in the bowl of a food processor and process until smooth. Strain, discarding solids, and measure out 1 cup of puree.

9. Place 1 cup raspberry puree in a medium saucepan along with 1/4 cup reserved beet puree and vanilla bean and seeds. Place saucepan over medium heat and bring to a boil.

10. In a small bowl, mix together sugar and apple pectin; add to saucepan and return to a boil. Strain sauce into a medium bowl and cover with plastic wrap. Transfer to refrigerator and chill for at least 2 hours or up to 2 days.

11. To serve: Using a 3 1/2-inch heart-shaped cutter, cut out six heart shapes from the cake. Using a 1 1/2-inch heat-shaped cutter, cut out the center of each 3 1/2-inch heart. Dust large hearts with confectioners' sugar; place one on each of six plates. Fill the center of each heart with raspberry-beet sauce and drizzle some of the sauce on the plate. Lean the smaller heart against the larger heart.

12. Using a soup spoon, scoop up a mounded spoonful of vanilla whipped cream; slide second soup spoon on top of first, with spoons pointing in opposite directions. Spoon whipped cream back and forth a few times, scraping it into a small, neat egg shape. Place on one of the plates. Repeat process five times until each plate has a scoop of whipped cream. Crush a rose petal over each plate and serve immediately.