In a stock pot, heat the oil over medium-high heat and cook the lamb in two batches until the meat is very deeply browned on at least two sides, 6 to 8 minutes per side. Reduce the heat if it threatens to burn. Transfer the lamb to a plate and pour off the oil in the pot.
Return the pot to medium-high heat and add the water, scraping the bottom of the pot to loosen any browned bits. Return the lamb and any accumulated juices to the pot and bring to a simmer. As it comes to a simmer, skim off any foam that rises to the surface with a ladle or large spoon.
Add the onion, carrot, celery, garlic, thyme, and salt. Cook, uncovered, at the barest possible simmer, stirring occasionally, for 2 hours.
Remove from the heat and let the stock rest for 15 minutes. Strain through a large fine-mesh strainer or a colander lined with a double layer of damp cheesecloth into a large bowl.
Fill a larger bowl or the sink with ice and water and nest the bowl of stock in it. Stir regularly until the stock has cooled.
Transfer to airtight containers and refrigerate for up to 3 days or freeze for up to 3 months.
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