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Step 1
In a medium bowl, cover rice with at least 2 inches cold water and soak for 1 hour. Drain well.
Step 2
Meanwhile, in a large Dutch oven or similarly sized pot, combine chicken, whole garlic cloves, ginger, and scallion whites with 2 quarts (2L) cold water. Bring to a simmer, then cover and cook at a gentle simmer until thickest part of breast close to bone registers 150°F (66°C) on an instant-read thermometer and joint between thighs and body registers at least 175°F (80°C), about 45 minutes.
Step 3
Carefully transfer chicken to work surface and let stand until cool enough to handle. Set a fine-mesh strainer over a medium heatproof bowl and strain broth through it; discard solids.
Step 4
In a 3-quart saucier or saucepan, heat sesame oil over medium heat until shimmering. Add drained rice and cook, stirring frequently, until a sticky film develops on bottom of pot, about 3 minutes. Add carrot, onion, and zucchini and cook, stirring frequently, until vegetables have softened slightly and rice grains are slightly translucent around their edges, about 2 minutes.
Step 5
Add 6 cups (1.4L) strained broth and stir to scrape up any starchy coating on bottom of pan. Bring to a rapid simmer, then cook, stirring occasionally to make sure nothing sticks to bottom of pan, until rice grains are plump and tender and liquid is thickened, about 30 minutes.
Step 6
Meanwhile, shred chicken meat, discarding skin and bones.
Step 7
Add two-thirds of both the chicken meat and scallion greens to juk along with remaining 2 cloves minced garlic and simmer until chicken is warmed through; if juk is thicker than you'd like, thin as desired with remaining broth (or reserve extra broth for another use). Season juk with salt to taste.
Step 8
Ladle juk into individual serving bowls, garnish with remaining shredded chicken and scallion greens, sprinkle with sesame seeds, and serve.