Place celery root and half-and-half in a medium saucepan over medium heat. Place thyme and garlic in a piece of cheesecloth; tie with kitchen twine to enclose. Add to saucepan.
When liquid comes to a boil, immediately reduce heat to low and simmer until celery root is tender, 20 to 25 minutes.
Remove and discard cheesecloth packet. Strain, reserving both celery root and cooking liquid. Transfer celery root to jar of blender and blend, incorporating enough of the cooking liquid to make smooth but not loose. With the blender running, add butter. Season with salt. Use immediately or cool completely in an ice-water bath. Transfer to an airtight container and keep refrigerated until ready to use.