Preheat oven to 400˚ F. Remove broccoli florets from the stems and cut into small pieces. Peel the stems, discarding the outside, then dice stems and set aside. Toss with 1/2 tablespoon olive oil and spread into a single layer on a sheet tray. Roast until broccoli is tender and starting to brown, 20 to 25 minutes.
While the broccoli is roasting, heat olive oil over medium low in a medium stock pot. Add in minced shallots. Cook for roughly 4 minutes then add in the ginger, cooking for one more minute. Stir in the broccoli stems followed by 3 cups of broth, lemon zest, juice, and salt. Bring to a boil, reduce to a simmer, and cook until stems are tender and roasted broccoli is done.
Add in the roasted broccoli, reserving a few smaller pieces for topping the soup. Transfer to a blender puree the soup. You will want a solid blender for the job (high-powered if you have it). If soup is too thick, add enough vegetable broth to thin it down to a good consistency.
Transfer the soup back to the pan if using a blender and add in the coconut milk. Heat the soup over low until ready to serve.
Divide soup into two bowls and top with the saved roasted broccoli, sesame seeds, scallions, and extra coconut milk.
+ make sure the stems are cooked through and if you blend the soup and find the texture not smooth, pass the soup through a sieve before returning to the pot.