Prep the leeks properly. Trim off the dark green tops and root ends. Slice the leeks lengthwise, then into thin half-moons. Rinse thoroughly in a bowl of water to remove grit.
Lift out and drain well.
Sauté the aromatics. In a large pot or Dutch oven, warm the butter and olive oil over medium heat. Add leeks, carrot, and celery with a pinch of salt. Cook 7–9 minutes, stirring often, until soft and fragrant but not browned.
Add garlic and thyme. Stir in garlic and thyme.
Cook 30–60 seconds until the garlic smells toasty.
Simmer the potatoes. Add potatoes, broth, bay leaf, and 1/2 teaspoon salt. Bring to a boil, then reduce to a gentle simmer. Cook 12–15 minutes, until the potatoes are very tender.
Blend for creaminess. Remove the bay leaf and thyme stems.
Use an immersion blender to puree until smooth. If you prefer some texture, blend only half the soup, or blend and then stir in a few reserved potato chunks.
Cook the orzo in the soup. Return the pot to a low simmer. Stir in the orzo and cook 8–10 minutes, stirring often so it doesn’t stick.
If the soup thickens too much, splash in more broth or hot water.
Finish with cream and lemon. Lower the heat and stir in the cream and lemon juice. Taste and adjust salt and pepper. The lemon should lift the flavor without making it tart.
Serve with a fresh finish. Ladle into bowls and top with chopped chives or parsley.
Add a twist of pepper, a drizzle of olive oil, or a sprinkle of Parmesan if you like.