Put peas into a medium-size saucepan with 2 cups chicken broth. 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 and set aside.
Put peas into 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 purée with the texture of loose mashed potatoes. Set aside. (If using frozen peas, you may want to strain the purée; freezing makes the skins tough.)
For the sauce, peel and cut carrots into 2-inch pieces; put in a saucepan with remaining 2 cups broth. Bring to a boil; simmer until tender.
Put carrots and broth in a blender with crème fraîche. Purée until very smooth. Add salt and pepper to taste.
To finish, heat olive oil in a nonstick sauté pan on medium-high setting. 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 purée; top with carrot sauce.