To prepare the broccoli, cut the thick stalks off the broccoli crowns. Cut the dry end off the stalk and peel the thick stalk with a vegetable peeler or paring knife. Cut the stalks into small chunks and set aside. Cut the florets off the thicker stems and put the small florets in a bowl and add the stems to the previously cut up stalk. The broccoli florets will go in last so keep them separated from the rest of the broccoli stalks and stems.
Cut bacon into paper clip size pieces and cook over a medium heat in a large pot (at least 3 or 4 quarts).
When bacon is crisp, remove it from the pot and set it aside on a plate for garnish later.
Add the chopped onion and garlic and cook over a medium low heat for 3-4 minutes.
Peel and cube the sweet potatoes and add them to the pot along with the broccoli stalks, stems, and the chicken broth. Stir with a wooden spatula and use it to get any bits off the bottom of the pot that are stuck.
Add half a teaspoon of salt (or less if you prefer less seasoning) and 1/4 tsp pepper, stir and cover the pot with the lid and let it simmer for 15 minutes or until the broccoli stalk pieces and sweet potato are very soft.
Use an immersion blender to puree the soup. You could also use a blender, working in batches and pouring the blended soup into a large bowl as you ladle out the soup from the pot before transferring the pureed soup back into the pot when it has all been pureed with the blender.
Once the soup is pureed smooth, mix the corn starch with 1 cup of water and whisk the corn starch slurry into the soup. Bring the soup to a simmer again.
Whisk in the cheddar cheese and keep whisking as it melts and combines with the soup.
Whisk in the cream (you can leave it out or use half and half cream if you prefer) and check to see if you need to add any more salt or pepper.
Bring the soup back to a steady simmer and add in all the broccoli florets. Simmer for 5 minutes, stirring occasionally to make sure that the thickened soup does form a film on the bottom of the pot. You can either add all the reserved bacon now or use it as a garnish when serving.
Serve hot with a nice big piece of crusty bread.