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.