In a small pot, combine the preserves, stock, Tamari and ginger, and bring to a low simmer.
In another pot, parboil the broccolini for 3 minutes in boiling, salted water then cool in an ice bath and drain; pat dry.
Heat a large, round-bottom pan or wok over high heat. Add 2 tablespoons oil, brown chicken and remove to plate. Add more oil and stir-fry the drained broccolini, peppers and onions for 2-3 minutes; add garlic, pepper, sauce and chicken back to the pan, and toss to combine. Finish with sesame oil and seeds. Serve over white or brown rice.