Place peas in a saucepan with 2 cups chicken broth (the remaining 2 cups will be used in the carrot sauce). Bring to a boil and simmer until just tender. To preserve green color, immediately scoop peas out of broth and plunge into ice water. Reserve broth; set aside.
Place peas in a blender or food processor. Add basil; add salt and pepper to taste. Start blending peas, adding enough broth as you go to create a smooth puree with the texture of loose mashed potatoes. Set aside. (If using frozen peas, you may want to strain the puree; freezing makes the skins tough.)
Discard excess broth.
For the sauce, peel and cut carrots into 2-inch pieces; place in a saucepan with remaining 2 cups of broth. Bring to a boil; simmer until tender. Place carrots and broth in a blender with creme fraiche. Puree until very smooth. Add salt and pepper to taste.
To finish, heat olive oil in a nonstick saute pan over medium-high heat. Sprinkle scallops with sugar and some salt and pepper. Sear each scallop until golden brown on each side, about 2 minutes per side. Serve a few scallops on the pea puree; top with carrot sauce.