In a small Dutch oven, melt butter over medium heat. Add leek, thyme, and garlic; cover and cook over medium-low heat until leek is softened but not browned, about 25 minutes.
Add potatoes, 3½ cups water, 1 teaspoon salt, and pepper; bring to a boil over medium-high heat.
Reduce heat to medium-low; partially cover and cook until tender, about 25 minutes. Remove from heat; let cool slightly.
Using an immersion blender, pulse soup until very smooth. Stir in half-and-half and remaining ½ teaspoon salt. Cook over medium-low heat until heated through. Garnish with leek, oil, pepper, and thyme, if desired.
Notes
Kitchen Tip: To use a regular blender, remove the center piece of lid to let steam escape and place a clean towel over opening in lid to avoid splatters. Avoid using a food processor.
Recipe by Southern Lady Magazine at https://southernladymagazine.com/creamy-leek-and-potato-soup-recipe/