Combine sugar and water in a saucepan and bring to a boil. Turn heat down and simmer until sugar dissolves, about 5 minutes.
Remove from heat. Add dandelion petals; stir to saturate petals with the syrup. Allow to cool slightly.
Pour syrup into a container and leave to infuse at least a few hours or overnight.
Before using, strain syrup through a cheesecloth, wringing out with your hands or pressing with the back of a wooden spoon to extract as much syrup as possible. You should end up with about ¾ cup of strained syrup.
For the dandelion cake:
Preheat oven to 350° F. Butter two 6-inch cake pans and line the bottoms with parchment paper.
Whisk together flour, dandelion sugar, baking powder, and salt in the bowl of a stand mixer.
Beat soft butter into flour mixture until butter is broken down into tiny pieces, about 3-4 minutes.
Whisk milk, egg whites, and vanilla bean seeds together. Pour milk mixture into butter-flour mixture and beat on medium speed until fluffy, about 3 minutes. Add remaining milk mixture and blend until just combined.
Distribute batter between pans and bake until lightly golden brown around the edges and starting to pull away from the sides of the pan, about 25-30 minutes. Allow cakes to cool; invert to remove from pans and remove parchment paper.
For the dandelion buttercream:
Pour dandelion syrup into a saucepan and cook over medium-low heat until it reaches 235 - 240° F.
Meanwhile, while syrup cooks, place egg whites into a clean bowl of a stand mixer fitted with the whisk attachment. Whip on high until soft peaks form. If soft peaks form before syrup is ready, turn mixture on low but keep them moving.
Once syrup is ready, remove it from heat and let it sit for about a minute, until bubbles subside. Pour it into egg whites in a very slow and gradual stream, with mixer on medium. Once all syrup is in, continue to whisk on medium for about 5 minutes, or until mixture has cooled slightly.
Start adding in soft butter, about a tablespoon at a time, until all the butter is in. Allow to beat a few more minutes until buttercream is silky and smooth.
Slice each cake in half horizontally to create 4 layers.
Place first layer of cake onto a cake plate. Add about 1/3 cup of buttercream and spread evenly onto cake, leaving a little bit of space around the edges.
Repeat with the next two layers of cake, adding a cake layer and spreading buttercream on top. Add final cake layer on top, pressing gently to adhere.
Frost cake in a very thin layer of buttercream on top and sides. Place in the refrigerator to chill for at least 20 minutes.
Remove cake from refrigerator and finish frosting with remaining buttercream. Sprinkle cake with additional dandelion sugar if desired.
Notes
Cake will keep in the refrigerator for about 4 days. For best flavor, allow cake to warm up to room temperature before serving.