1. Make the stock and chicken: Place the chicken in a large stockpot over medium heat. Pour in just enough cold water to cover the chicken (about 3 quarts); too much will make the broth taste weak. Place half an onion, 2 stalks of celery, and 2 tablespoons of salt in the stockpot. Bring to a boil, reduce heat to medium-low, and gently simmer partially covered for 1 hour 15 minutes, or until the chicken is done (the chicken is done when the drumstick easily pulls off the bird). As it cooks, skim any foam that rises to the surface, adding a little more water if necessary to keep the chicken covered while simmering. Once the chicken is done, remove it from the pot and let cool.  

2. While the chicken cooks, break the cauliflower florets from the stem. Dice the remaining onion, celery, carrots, and mince the garlic.

3. In a large skillet over medium heat, add the olive oil and swirl around the pan before adding the onion, celery, and carrots. Cook, stirring frequently, for 10 minutes. Reduce the heat to low and add the garlic, stir, and cook for an additional 5 minutes (do not let the garlic brown, the flavor will turn bitter). Turn the heat off, cover, and let sit until ready to prepare the soup. 

4. Strain the chicken stock and remove the cooked vegetables and any impurities; discard the cooked vegetables. Place 7 cups broth back in the pot over medium heat. Reserve any leftover broth for future use (refrigerate for up to a week, or freeze for extended time). Add the cauliflower florets to the broth and bring to a boil. Reduce heat to a simmer and cook covered until tender, 7-8 minutes. Blend until smooth with an immersion blender or transfer the cooked cauliflower in 2-3 batches to a blender and add just enough liquid to almost cover it—because the cauliflower and liquid are hot, fill the blender only halfway. When blending hot liquid, remove the center piece from the blender lid, and place a clean folded dishtowel over the opening; turn to the lowest setting and work your way up to blend until smooth, using caution to avoid splashing. Pour back into the stockpot and transfer more cauliflower florets to the blender, repeating the process until the cauliflower is puréed with the broth. 

5. Add the seasonings, half and half, peas, and the sautéed onion, celery, carrots, and garlic. Stir to combine, and if necessary, add additional salt to taste.  

6. Remove the chicken meat from the bones (set aside skin and bones for making another broth, or discard). Roughly chop the meat and add to the pot. Simmer over medium-low heat for an additional 10 minutes; if needed, add additional broth to reach desired consistency. Serve with optional toppings.