Melt butter in a large pot over medium heat. Cook and stir leeks, garlic, salt, and pepper in the hot butter until leeks are tender and lightly browned, about 15 minutes.

Stir chicken broth and cornstarch together in a bowl until cornstarch is dissolved; pour over the leek mixture. Add potatoes to the broth mixture and bring to a boil; season with onion powder and garlic powder. Reduce heat, add half-and-half, and simmer until potatoes are tender, about 30 minutes.

Remove pot from heat and blend soup using a hand blender until smooth; season with salt and pepper.