Ideally, go out after a rain when the soil is soft. Dig up the largest clusters of dandelions you can, then trim the "hearts" off and save for another purpose, then break the roots into manageable pieces.
Scrub the roots with an abrasive cleaning pad in warm water. Trim very dirty areas with a paring knife. Remove and small root threads to make them easier to clean.
Cut the roots into small pieces and immerse in warm water again just to be safe, agitating them to remove grit. If you'll use them for deserts, inspect them closely as they must be pristinely clean.
Dehydrate the roots for 24 hours or until bone dry at 145F or a similar temperature. From here the dried roots can be stored in a jar in the pantry for a very long time.
Roast the Dandelion or Chicory Roots
Note that you can use fresh, cleaned roots to do this as well, without dehydrating first.
Put the dried roots in a cast iron pan and bake at 300F for 1 hour.
Cool the roots for a few minutes, then grind to a powder in a spice grinder. For maximum caramelization, or if you're using wild roots which seem to take longer to brown, toast the ground powder in the oven for another 15-20 minutes.
Dandelion or Chicory Coffee
Combine the ground dandelion roots and water and bring to a boil, turn the heat to a simmer and cook for 10 minutes, then serve as you would coffee.
New Orleans Coffee
Combine the ground coffee and dandelion roots with the water in a pot and bring to a boil. Turn the heat to a simmer and cook 10 minutes, then strain.
Combine the strained coffee with the milk and serve.
Video
Notes
Many different roots can be used here alone or in combination with dandelion and chicory root. Wild parsnips and burdock would be my first choice to experiment with.