Place the carrots and stock in a large skillet Set over low heat, cover, and cook, stirring occasionally, for 15 minutes or until the carrots are crisp-tender.
While the carrots cook, place the Jerusalem artichokes, ginger, garlic, and milk in a medium-size saucepan.
Bring to a boil, then reduce heat, cover, and simmer for 15 minutes.
Sprinkle the flour over the carrots in the skillet and stir until the carrots are coated.
Add the Jerusalem artichokes and milk mixture to the carrots and stir.
Simmer for 15 to 20 minutes.
Place the soup in a blender.
Process on low, then medium speed until smooth, stopping and scraping down the sides as necessary.
Put the soup through a strainer and press out liquid with the back of a wooden spoon to thin the soup.
Return to the skillet or a saucepan and heat through.
Garnish with parsley, if desired.