Rinse shrimp shells under cold running water to remove any debris.

Heat a 6-quart stockpot over medium-high heat and dry-roast shrimp shells, stirring for 2 to 3 minutes until they turn bright pink.

Add halved onion, chopped celery, chopped carrots, and smashed garlic to the pot.

Pour in 6 cups cold water to cover the shells and vegetables by about 1 inch.

Bring the mixture to a rolling boil over high heat, then skim off any foam that rises to the surface.

Reduce heat to low and add thyme sprigs, bay leaves, and peppercorns; simmer gently with occasional bubbles.

Let the stock simmer undisturbed for 30 to 40 minutes, tasting at 30 minutes for flavor.

Remove from heat and cool for 5 minutes, then strain through a fine mesh strainer lined with cheesecloth into a clean container.

Cool to room temperature, season with salt to taste, then refrigerate for up to 4 days or freeze for up to 3 months.