Bring a large pot of water to a boil over high heat.
Lower the shrimp into the water, and cook for about 1 1/2 minutes, or until just cooked.
Drain the shrimp, let cool, and remove the tails and shells. Set the cooked shrimp aside, and place the shells into a mortar and pestle, or food processor, and roughly chop (or crush).
Heat a medium saute pan over medium heat. Add the 1/4 cup butter and, when it has melted, add the onion, carrot, and celery and cook, stirring frequently, until the onion is tender and lightly golden.
Add the crushed shrimp tails and shells and saute for another 5 minutes over low heat.
Add the brandy and simmer until the brandy has cooked away, about 3 minutes.
Add the Marsala and cook until the Marsala has cooked away, 3 minutes more.
Add the tomato paste, bay leaves, and thyme; cook, stirring frequently, until the mixture has a rich, rusty color and a sweet aroma, another 5 minutes.
Add water to just barely cover the shells and add the crumbled amaretti cookies, and stir to combine and simmer over very low heat for 30 minutes.
Strain the stock through a fine-mesh sieve, pressing the solids well to extract as much liquid as possible. You should have at least 2 cups; reserve any additional stock to use in other dishes.