Roughly chop the white and light green parts of the leeks, placing them into a medium glass bowl. Cover the leeks with cold water and allow them to soak for 5 minutes, until the grit falls to the bottom. Remove the leeks to a towel and squeeze out as much water as you can.
Melt 1 tbsp Earth Balance and combine with rice milk in a small bowl and set aside.*
Melt the remaining 4 tbsp of Earth Balance over medium in a large stockpot. Add the leeks and garlic and cook until fragrant and soft but not brown, about 8 minutes.
Add the potatoes, broth, bay leaf, dried thyme, salt and a couple of pinches of black pepper and bring the contents to a boil. Cover, turn the heat down to medium-low and gently simmer for about 15 minutes, or until the potatoes break easily when pierced with a fork.
Pull out the bay leaf, then puree the soup with an immersion blender right in the pot, or in batches in a blender or VitaMix; return to the pot. Add the oat milk and Earth Balance mixture, stir to combine and heat through. Taste and add more salt and pepper if needed.
Ladle into bowls and garnish with chives or more dried thyme.
Notes
*Note: If your family does not have nut allergies, you can substitute one cup of almond or cashew milk for the rice milk and Earth Balance mixture.