Sauté: Heat the olive oil (or a splash of water) in a large pot over medium heat. Sauté the carrots and celery for about 6–8 minutes until softened.
Simmer: Add the potatoes, beans, vegetable broth, thyme, and bay leaf. Bring to a boil, then reduce heat and simmer uncovered for 15–20 minutes, or until the potatoes are fork-tender.
Incorporate Spinach: Turn off the heat entirely. Add the fresh spinach to the pot and stir gently. The residual heat from the broth will wilt the spinach almost instantly without making it mushy.
Serve: Once the spinach is bright green and wilted (usually about 30 seconds), remove the bay leaf and serve.