Prep: Peel and slice the garlic; dice the onion; peel and cube the potatoes; cube stale bread if using. Measure stock and have herbs ready.
Warm the pot: Heat a 3–5 qt heavy-bottomed pot or Dutch oven over low–medium-low heat. Add 2–3 tbsp olive oil and warm until it shimmers (do not let smoke).
Sweat garlic & onion: Add garlic and onion to the oil. Cook gently over low–medium-low heat, stirring frequently, for 10–12 minutes until soft and translucent but not browned. The goal is to mellow the garlic, not crisp it.
Add potatoes & herbs: Add cubed potatoes, bay leaf, and thyme; stir to coat and cook 3–4 minutes to wake up the potatoes.
Deglaze (optional) & add stock: Pour in the white wine (if using) and reduce for 1 minute. Add stock and water, bring to a gentle simmer, cover partially, and simmer 20–25 minutes until potatoes are very tender.
Rustic: Remove bay leaf; mash about one-third of the soup with a potato masher and stir back to create body while leaving bite.
Silky: Remove bay leaf and use an immersion blender in the pot (or puree in batches in a blender) until smooth and velvety. For ultra-smooth, strain through a fine mesh sieve.
Finish & season: Stir in sherry vinegar or lemon juice. Taste and adjust salt and pepper. For extra richness, swirl in 1 tbsp butter or an additional tbsp of olive oil.
Optional gratinée: Preheat broiler. Ladle soup into oven-safe bowls, top with a toasted slice of baguette or crouton, sprinkle with grated Gruyère or Parmesan, and broil 2–3 minutes until bubbly and golden. Serve immediately.
Garnish & serve: Drizzle with a little extra virgin olive oil and scatter chopped parsley or chives on top. Serve hot.