In a large soup pot, heat the olive oil or butter over medium heat.
Add the onion, garlic, celery, and carrot. Sauté for 5–7 minutes until softened and fragrant.
Stir in the potatoes, corn, thyme, bay leaf, salt, and pepper. Mix to coat.
Pour in the vegetable broth and bring the soup to a boil.
Reduce heat and simmer for 15–20 minutes, or until potatoes are tender.
Remove the bay leaf. Use an immersion blender to partially blend the soup (or transfer half to a blender and return).
Stir in the half-and-half or dairy-free cream. Simmer gently for another 5 minutes—don’t boil.
Taste and adjust seasoning.
Serve hot, garnished with herbs or your favorite toppings.