Fill a deep frying pan halfway full with water and bring to a boil. Add asparagus to the boiling water and add salt and olive oil; boil for 6 minutes. Remove asparagus to a sheet pan or plate to cool, and place in the refrigerator for 5 to 10 minutes. Then chop into bite-size pieces, perfect for dipping.
Melt butter in a skillet over medium heat. Add garlic and saute for about 1 minute. Add cream and simmer until reduced by half. Season with salt and pepper. Add cheese, bit by bit, whisking as you go. As 1 piece melts, add another, until no cheese remains. Then add thyme and stir to combine. Serve with cubes of day-old bread and the cooled asparagus pieces.