Bring the barley and water to a boil in a saucepan over high heat.
Cover, reduce heat to low, and simmer until the barley is tender, about 30 minutes.
Drain and set aside.
Melt the butter in a large pot over medium heat.
Stir the onion and celery into the melted butter, place a cover on the pot, and cook until tender, about 5 minutes.
Add about half the garlic.
Season with the black pepper, red pepper flakes, sea salt, and nutmeg.
Return the cover to the pot and cook another 5 minutes.
Stir the broccoli, vegetable stock, and remaining garlic into the mixture and bring to a boil.
Return the cover to the pot, reduce heat to medium, and cook another 15 minutes, stirring once.
Remove the pot from heat.
Pour the soup into a blender, filling the pitcher no more than halfway.
Hold the blender lid with a folded kitchen towel and carefully start the blender, using a few quick pulses to get the soup moving before leaving it on to puree.
Puree in batches until smooth and pour into a clean pot.
Alternately, you can puree the soup with a stick blender directly in the pot.
Return the soup to medium-low heat.
Stir the coconut milk, 2% milk, and barley into the pureed soup; cover and cook until hot, 5 to 8 minutes.
Ladle into bowls and top with Asiago cheese and parsley to serve.