Step 1Shell the shrimp. Place the heads and shells in a medium pot and cover with cold water. Bring to a boil, reduce the heat and simmer 15 minutes. Strain the broth and reserve.
Step 2Heat a 4-quart saucepan over medium-high heat. Add 3 tablespoons butter, the onions and bay leaf. Cook, stirring constantly, until the onions start to brown, about 5 to 6 minutes.
Step 3Reduce the heat to medium and add the squash. Cook, stirring occasionally, until the squash begins to soften and turn brown, 7 to 8 minutes.
Step 4Reduce the heat to low and add 1 tablespoon butter and the peeled shrimp. Cook, stirring occasionally, until the shrimps turn pink, 3 to 4 minutes.
Step 5Add the salt, white pepper, cayenne, basil, thyme and garlic. Cook 3 to 4 minutes, stirring constantly and scraping the bottom and sides of the pan with a metal spoon. (This procedure will intensify the taste of the soup.) Add 1 1/2 cups reserved shrimp stock and simmer 2 to 3 minutes.
Step 6Remove the pot from the heat. Discard the bay leaf. Transfer the contents to a food processor and puree until smooth. Return the puree to a clean saucepan and add the cream and half-and-half. Bring to a boil, lower the heat and simmer 2 to 3 minutes. Just before serving, add the fresh basil and remaining 1 tablespoon butter. Top each serving with several sauteed shrimp.