Place a Dutch oven over medium heat, add in the butter and oil, till it's melted and mixed. Add in the leeks, celery, and potatoes. Cook while stirring occasionally for about 5-7 minutes, or until the leeks are tender.

Stir in the garlic, and cook for a few minutes until fragrant. Stir in the chicken broth, thyme, salt, and pepper, and cook, uncovered, until the potatoes are tender, about 20 minutes.

Remove the thyme sprigs, and transfer about 4 cups of the soup to a blender and puree until smooth. Transfer the pureed soup back into the Dutch oven, and stir in the heavy cream. Reduce the heat to medium-low, and cook while stirring until the soup is slightly thickened, for about 12-15 minutes.

Taste the soup, and see if you need additional salt or pepper.

Viola! Your potato leek soup is ready, garnish with finely chopped chives when serving.