Heat the olive oil in a large pot over medium heat. Add the diced onion and cook until it softens, about 4 minutes. Stir in the minced garlic and cook for 1 minute more.
Add the broccoli florets, diced potato, and low-sodium vegetable broth. Bring the mixture to a boil, then reduce the heat and simmer until the potato is tender, about 15 minutes.
Remove the pot from the heat. Use an immersion blender to puree the soup until smooth, or transfer it in batches to a countertop blender. Return the soup to low heat and stir in the unsweetened almond milk and black pepper. Heat through for 2 to 3 minutes without boiling.
Remove from heat and stir in the lemon zest. Ladle into bowls and top each serving with chopped fresh chives.