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1
Using a cleaver or kitchen scissors, break up the carcass into several smaller pieces so that they will fit in an even layer in the bottom of a large pot or Dutch oven; set aside.
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2
Heat the oil in the pot over medium-high heat until shimmering.
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3
Add the carcass pieces and cook, turning occasionally, until lightly browned all over, about 8 minutes.
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4
Add the remaining ingredients and stir to combine, scraping up any browned bits from the bottom of the pot with a wooden spoon.
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5
Bring to a simmer (do not let the stock come to a boil).
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6
Reduce the heat to low and continue to simmer, occasionally skimming any scum off the surface of the stock using a large spoon.
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7
Cook, adjusting the heat as necessary to keep the stock at a simmer, until the flavors have developed, about 1 1/2 hours.
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8
Remove and discard any large pieces of carcass.
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9
Set a fine-mesh strainer over a 2-quart saucepan and pour the stock through the strainer (you should have about 6 cups).
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10
Discard the contents of the strainer.
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11
(At this point, the stock can be cooled to room temperature and refrigerated in a container with a tightfitting lid for up to 3 days or frozen for up to 1 month.)
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12
Bring the stock to a simmer over medium-high heat.
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13
Add the carrots, celery, onion, measured salt, and thyme, season with pepper, and stir to combine.
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14
Return to a simmer, reduce the heat to medium low, and simmer until the vegetables are tender, about 20 minutes.
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15
Meanwhile, bring a medium saucepan of heavily salted water to a boil over high heat.
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16
Add the noodles and cook according to the package directions until al dente.
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17
Drain in a colander.
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18
While the vegetables and noodles cook, shred the reserved chicken meat from the carcass into bite-size pieces.
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19
Reserve 1 1/2 cups for the soup; save the rest for another use.
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20
When the vegetables are tender, add the drained noodles and shredded chicken, stir to combine, and return the soup to a simmer.
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21
Cook until the flavors meld, about 5 minutes more.
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22
Taste and season with salt and pepper as needed.
