Peel and dice the potatoes into roughly 1-inch chunks. Set aside.
Thinly slice the onion and mince the garlic cloves to prepare for sautéing.
Heat the olive oil in a large heavy-bottomed pot over medium heat until it shimmers and begins to smell fragrant.
Add the sliced onion and minced garlic to the pot. Cook, stirring often, until the onions turn translucent and fragrant, about 5 minutes.
Add the diced potatoes to the pot, stirring gently to coat them with the aromatic oil and soften slightly, about 2 minutes.
Pour in the vegetable broth, bringing the mixture to a boil over high heat.
Reduce the heat to low, add the bay leaf or thyme, and let the soup simmer gently for 20-25 minutes until the potatoes are very tender and breaking apart.
Remove the bay leaf or thyme sprig, then use an immersion blender directly in the pot, or carefully transfer the soup in batches to a blender, blending until smooth and velvety.
Taste the soup and season with salt and freshly ground pepper, adjusting to your preference. If desired, stir in a splash more broth or water for your preferred consistency.
Let the soup rest off the heat for 5 minutes, allowing flavors to deepen and settle.
Spoon the warm, creamy soup into bowls, and garnish with a drizzle of olive oil or fresh herbs if you like. Serve hot and enjoy the comforting, rustic flavors.
