Slice the root ends of the leeks off, then slice the white and light green parts into ½-inch rounds. (Save the dark green parts for making stock!) Place sliced leeks in a medium bowl and cover with water. Swish the leeks around, separating the rings to make sure all the grit is cleaned off. Remove cleaned leeks onto a kitchen to drain. Discard dirty water.
Heat butter or oil in a large pot over medium heat. Add leeks and onions and saute for 10 minutes, until beginning to soften.
Add celery and saute for 10 minutes, until softened and vegetables are beginning to caramelize. This is key to a really flavorful soup.
Add garlic and thyme and saute for 1 minute, until fragrant.
Increase heat to high and add wine. Let cook for 3-4 minutes, until the wine has reduced, scraping up any flavorful bits from the bottom of the pan.
Add potatoes, stock, bay leaf, salt, and pepper to the pot and stir to combine. Bring to a boil and then reduce heat to low and cover. Allow soup to simmer for 10-15 minutes until potatoes are tender.
Remove bay leaf and stir in peas. Return soup to simmer and let cook for 5 minutes.
Taste and adjust seasoning as needed before serving. Serve with a drizzle of extra virgin olive oil and a sprinkle of parmesan cheese.