Make broth: Although you could use canned broth instead, you can make your own with the bones from the same chicken that supplies meat for the congee. Separate wings from chicken and pull meat from bones; discard skin. Set bones and wings aside and shred meat into bite-size pieces.
Heat oil in a medium pot over medium heat. Add garlic, chopped onion, and carrot and cook until softened, about 5 minutes. Add wings, bones, peppercorns, and bay leaves, along with enough water to just cover bones (4 to 5 cups). Bring to a boil, then lower the heat to a slow simmer. Skim the pot after 15 minutes and occasionally throughout cooking. Simmer stock 2 to 3 hours, then strain through a fine-mesh strainer.
Meanwhile, make porridge base: Bring 10 cups water to a boil. Whisk in jasmine and sticky rice. Reduce heat to low and simmer, covered, stirring every 5 minutes, until rice looks like loose cooked oatmeal, about 45 minutes. Set aside 6 cups.
Bring a medium pot of water to a boil and carefully add eggs. Lower heat to a simmer and cook eggs 8 minutes, then plunge into an ice water bath. When they're cool, peel.
Bring porridge base and 2 cups chicken stock to a boil. Stir in 2 cups chicken meat and reduce heat to a simmer. Season with salt and pepper.
Break or slice eggs in half. Divide congee among four large soup bowls. Top each with sprouts, green onions, cilantro, egg, and a couple of slices of savory Chinese doughnuts. Add a few slivers of ginger last to keep it from cooking in the congee. Nestle a lime wedge next to each bowl and serve.
Make ahead: Porridge base, up to 5 days, chilled airtight (it thickens in the fridge and can be used that way, but if you like it looser, add a little water).
*Broken rice, a by-product of processing and transporting rice, is stickier and softer than whole-grain rice. Find it at Vietnamese markets.