Heat the oil in a large pot over medium-high heat. Add the onions and garlic, cover, and cook until golden, stirring occasionally, about 8 minutes. Add the water, carrots, celery, mushroom stems, parsley stems, thyme, salt, black pepper, and bay leaves. Bring to a boil and decrease the heat to medium. Cook until the vegetables are tender and the broth is well flavored, about 30 minutes. Strain the broth into a large bowl using a colander. Press on the vegetables in the colander to release as much liquid as possible. Discard or compost the vegetables. Strain the broth using a fine-mesh strainer.