Wash and peel potatoes then cut into 3/4-inch cubes.
Peel the outer layer of leeks and rinse to remove dirt. Trim the ends of the leeks, removing the very bottom of the white portion of the leek with the roots and tips of the green leaves.
Cut off 3-4 inches from the bottom, removing the white leek bulb from the leaves. Cut the green leaf section in half lengthwise, then slice. Cut the white bulb in half and chop.
Heat olive oil and butter in a large pan. Add leeks and 1/2 teaspoon salt, cover, and cook until leeks are soft about 10 minutes.
Add potatoes and garlic along with 1 teaspoon salt. Stir to combine and cook for 5 minutes to allow flavors to meld.
Add vegetable broth and cook until potatoes are fork-tender, about 10-15 minutes.
Transfer 3 cups of soup (including broth, leeks, and potatoes) to a blender. Blend until smooth.
Return the blended soup to the pan with remaining soup.
Cook for 20 minutes, taste, and add additional salt and pepper as needed.
Serve hot with your favorite crusty bread or a roast beef or ribeye sandwich.