1. Heat the oil in a large pot over high heat until just smoking. Add the shrimp and sear until the shells are bright orange on both sides, about 3 minutes. Add the vegetables and tomato paste, turn the heat to medium-low, and cook, stirring often, until the vegetables are softened, about 5 minutes.

2. Stir in the water and bring to a boil. Lower the heat slightly and boil for 15 minutes, skimming off the foam as it rises. Remove from the heat and pour the mixture into a food processor. Pulse until the shrimp is coarsely chopped. Pour back into the pot and let stand for 10 minutes.

3. Strain through a fine-mesh sieve, pressing firmly on the solids to extract as much of the flavorful liquid as possible. Store, tightly covered, in the refrigerator for up to 3 days, or in the freezer for up to 2 months.