Cook the onion in the vegetable oil in a medium saucepan over low heat, covered, stirring occasionally, until soft, about 5 minutes. Then raise the heat to medium and cook, stirring often to prevent burning, until the onion is “caramelized” and a deep golden brown, about 15 minutes. Remove and set aside.
Put the beef in the same pan and sear over high heat until crusty and browned. Then add the mushroom trimmings and browned onions and cook together 5 minutes longer. Pour the chicken stock over the beef mixture, bring to a boil, and immediately lower to a bare simmer. Skim off any scum or fat that rises to the surface and cook for 1½ hours.
Strain through a fine-mesh sieve and discard the cooked beef and onion. Let the beef stock cool, then refrigerate or freeze.
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