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1
Heat the oil in a large wok or wide heavy pot over high heat. Add the chicken pieces in a single layer and sear without disturbing until the skin is golden and releases easily from the surface, because a piece that is forced too early will tear and lose the crust that carries flavour through the whole braise — work in batches if necessary. The pieces should colour, not steam.
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2
Push the chicken to the sides of the wok. In the cleared centre, fry the dried chillies, cumin seeds, and Sichuan peppercorns for thirty to forty seconds, so the dry heat coaxes the full, assertive fragrance from them that no amount of simmering in liquid can replicate — this is where the dish lives or dies; insufficient cumin at this stage cannot be corrected later. The kitchen should carry an identifiably Xinjiang, almost smoky spice scent before you move on.
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3
Add the tomato paste and stir to coat the chicken pieces. Cook for one minute until the paste darkens slightly from bright red to a deeper terracotta, because that extra minute drives off the raw, acidic edge and leaves only the rounded, savour depth you want in the braise. The sharp, canned smell will give way to something quieter and sweeter — that is the moment to proceed.
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4
Pour in the soy sauce and water or stock so that the liquid comes roughly halfway up the chicken pieces, then bring to a boil and reduce to a vigorous simmer. Add the potato chunks at this point so they have the full braising time to drink in the spiced sauce rather than merely cooking through in plain water. The broth should be sufficient to eventually saturate the noodles — add more water if the pot looks scant.
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5
Cover and braise until the potato chunks are completely tender and have absorbed the braising flavour throughout, checking by pressing a chunk against the side of the pot — it should yield without resistance but hold its shape entirely. Keeping the lid on through this stage traps the steam and keeps the liquid level honest, so you are not constantly compensating for evaporation as the potatoes cook.
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6
Add the red and green bell pepper pieces and cook uncovered for three to four minutes, leaving the lid off so that any excess liquid reduces a little and the sauce tightens around the other ingredients rather than turning watery on the plate. The peppers should soften slightly at the edges while retaining visible colour and a faint crunch at the centre.
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7
Transfer everything to the largest plate or shallow bowl available. Lay the cooked, drained noodles directly over the chicken and potatoes just before bringing to the table so that they begin soaking up the braising sauce in front of the diners rather than turning sodden in the kitchen. The sauce must be substantial enough to soak them through — if it looks thin, reduce it by a minute or two before plating.