Heat the olive oil in a medium Dutch oven over medium heat. Add the celery and onion and cook until they begin to wilt, about 4 minutes. Add the celery root and toss to coat in the oil. Add the thyme and season with 1 teaspoon salt and a generous grinding of black pepper. Cook, stirring, until the celery root beings to wilt, about 4 minutes.
Add the chicken broth, bay leaf and 2 cups water. Bring to a simmer and cook until the vegetables are tender, about 30 minutes. Add the oat milk and simmer to incorporate, 3 to 4 minutes.
Remove the bay leaf and blend the soup in a blender, in batches, until silky and smooth. Return the soup to the pot to heat through. Season with salt and pepper, if necessary. Serve in bowls with a drizzle of olive oil and some chopped chives.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.