Heat the olive oil in a large, heavy enamel soup pot over moderate heat. Slice fennel to remove fronds and ends then transfer to the pot and saute until transparent, stirring frequently.
Add carrots and beef or vegetable stock. Bring to a boil. Cover, reduce the heat slightly. Simmer gently, partially covered, until carrots are tender.
Remove the soup from the heat and spoon into a food processor. Process until smooth, then stir puree back into the pot. (NOTE: If you won't want to deal with the messy transition of placing your hot soup inside a regular blender, you can use a hand immersion blender instead. Be sure the soup isn't boiling and has cooled a bit.)
Add water and bring to a simmer. Taste the soup, adding salt or pepper if necessary. Increase the heat and bring to a lively simmer for an additional 30 minutes.
Ladle hot soup into hot jars leaving 1-inch headspace. Use a de-bubbler to remove air bubbles. Clean jar rims by wiping with warm water or distilled vinegar. Center lid on jar and adjust band to finger-tip-tight. Pleace the rack in a pressure canner containing 2 inches of simmering water (180 degrees Fahrenheit). Repeat until all jars are filled. *You must process at least 2 quart jars or 4 pint ars in the pressure canner at one time to ensure safe processing.
Place the lid on the canner and turn to the locked position. Adjust heat to medium-high. Vent steam for 10 minutes. Then, put a weighted gauge on the vent; bring pressure to 10 lbs (psi). Process quart jars for 50 minutes and pint jars for 40 minutes, adjusting for altitude, according to your pressure canners directions.
Turn off heat and cool the canner to zero pressure. Let stand for 5 more minutes before removing the lid. Let the jars cool in the canner for 10 minutes.
Remove jars from the canner; do not tighten bands if loose. Check lids for seal after 24 hours. A safely sealed jar lid will not flex up and down when the center is pressed. Label and store jars.