Cut the squid bodies on one side so they open up flat. Score with a sharp knife in a 1/4-inch crosshatch pattern. Slice into 1/2-inch strips. Cut any tentacles in half.
Cook the noodles in a pot of boiling salted water until tender, about 8 minutes. Drain and run under cold water until cool.
Stir together the broth, molasses and soy.
Heat the oil in a wok or large heavy skillet over high heat. When it just begins to smoke, add the bell peppers, 1 teaspoon salt and 1/2 teaspoon pepper and cook, stirring, until charred in places and crisp tender, about 3 minutes. Stir in the scallion whites, ginger and garlic and cook, stirring, until very fragrant, about 2 minutes. Add the squid and sauce to the wok and cook, stirring until the squid is just cooked, about 1 minute. Stir in the lime juice, and then toss the squid mixture with the noodles. Chill until cool, about 30 minutes or up to overnight.
Season with salt and pepper. Serve sprinkled with the cucumbers and scallion greens.
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