Warm the olive oil in a large soup pot over medium heat. Add the chopped onion and sauté gently for about 5 minutes, stirring occasionally, until soft and translucent without browning.
Add the diced potato to the pot along with the vegetable broth or water. Bring the mixture to a gentle boil, then lower the heat, cover the pot with a lid, and simmer for 10 minutes, or until the potato is fork-tender and fully cooked.
Stir in the green peas and let them simmer for approximately 4 minutes. Avoid overcooking to preserve their vibrant color and delicate, fresh flavor.
Switch off the heat and season the soup generously with salt (about one teaspoon, or to your taste), freshly cracked black pepper, and the mint leaves. Allow the soup to cool slightly before proceeding to blend.
If using an immersion blender, blend the soup directly in the pot until velvety smooth. If using a traditional blender, carefully transfer the soup in batches—do not exceed the maximum fill line to avoid overflow. If the consistency feels too thick, add a little extra broth or water to achieve your desired texture.
Return the puréed soup to the pot and reheat gently if needed. Taste once more and adjust the seasoning to your preference.
Serve the soup hot, garnished with a swirl of extra virgin olive oil, a handful of reserved peas, toasted pine nuts, and a crack of black pepper if desired. Enjoy each spoonful of this fresh and comforting green pea soup.