Peel and chop the potatoes into roughly 2 cm cubes, ensuring they are evenly sized for uniform cooking.
Heat a large pot over medium heat and add the olive oil, letting it shimmer as you prepare for sautéing.
Add the diced onion to the pot and cook, stirring occasionally, until it becomes translucent and fragrant, about 5 minutes.
Toss in the chopped potatoes, stirring to coat them with the onion and oil mixture, and cook for another 2 minutes.
Pour in the vegetable broth, ensuring it covers the potatoes, then bring the mixture to a gentle boil.
Reduce the heat to low, cover the pot, and simmer until the potatoes are very tender, about 15-20 minutes, with a gentle bubbling sound filling the kitchen.
Carefully remove the pot from heat, then use an immersion blender to blend the soup directly in the pot until it becomes smooth and velvety, or transfer in batches to a blender, blending carefully with hot liquids.
Stir in the plant-based milk, adjusting the consistency to your liking—add more if the soup is too thick, stirring until well incorporated.
Season the soup generously with salt and pepper, tasting and adjusting until it's perfectly balanced and flavorful.
Ladle the hot soup into bowls, garnish with chopped herbs or seasonings if desired, and serve with crusty bread for a cozy meal.