In a small mixing bowl, combine salt, sugar, and pepper. Firmly press the rub into the sliced ginger (extra spice rub may be saved to season soup, to taste).
Next, place the ginger into a sous vide pouch, add liquid smoke, and seal. Allow the ginger to cure for eight hours, or overnight for more flavor intensity.
Once cured, remove the ginger from the bag and reserve a few slices for garnish.
FOR THE CARROT SOUP:
Set the circulator to 181.4°F (83°C).
Combine the ginger and all ingredients except stock in a large sous vide pouch. Seal and cook for four hours or until vegetables are tender.
Once done, remove and transfer hot vegetables into a high-speed blender.
Gradually add the stock until the soup has a smooth consistency. While blending, season with reserved ginger-curing spices to taste.
Once the soup has reached the desired texture, strain through a chinois.
FOR THE WHIPPED CREAM:
Place all ingredients into a mixing bowl and whisk until stiff peaks form.
TO SERVE:
Serve in a soup bowl with a dollop of whipped cream and garnish with julienned smoked ginger and lime zest.