In a bowl, combine the cabbage, carrots, scallions, 1 tsp. soy sauce, 1 tsp. sugar, and a big pinch of salt. Set aside for 20 minutes, tossing occasionally.
In a small food processor, chop the garlic and ginger. Scrape the bowl with a spatula, add the 1/4 cup cilantro and chop thoroughly. Scrape the bowl, add the peanut butter, 1 Tbs. hot water, the lemon juice, 1 Tbs. oil, 3 Tbs. soy sauce, and 2-1/2 tsp. sugar. Pulse until well combined, scraping the bowl as needed.
In a large skillet over medium-high heat, heat 1 Tbs. oil. Season the chicken pieces with salt and put them in the hot pan (in batches, if necessary). Cook on one side until the edges are white (1 to 2 minutes); turn and cook until just firm, another 1 to 2 minutes. Transfer to a cutting board and stack them in one or two piles; let rest for 3 to 4 minutes. Slice the chicken into generous 1/4-inch strips and put them in a large bowl. Squeeze the cabbage mixture well and add it to the chicken, pour in the dressing, and toss well to combine. Divide among four plates and garnish with the remaining cilantro and the toasted almonds or peanuts.