Roughly chop the leeks, carrots, celery, and garlic. Peel the potatoes, cut them in half, and slice into thin wedges.
Melt the butter in a large saucepan over medium heat and add the leek, carrot, celery, garlic, bay leaf, and thyme. Cook over moderate heat, stirring occasionally, for about 5 to 10 minutes.
Add the potatoes and the broth. Give the pot a good stir, cover it with a lid, and simmer the mixture for about 20 minutes or until the potatoes are very tender.
Let the mixture stand until it is cool enough to blend. Discard the bay leaf and puree the soup with an immersion blender until it is very smooth.
Place the blended vegetable mixture back on the stove over medium low heat and stir in the half-and-half.
When the mixture is heated (but not boiling), stir in the crumbled Stilton until the cheese is melted and smooth.
Mix in the tawny port and add salt and pepper to taste.
Serve garnished with fresh chopped chives.