Slice the cabbage in half lengthwise. Position one half cut side up and slice it across the middle at the point where the ruffled, leafy top gives way to white stem. Remove the core, slice the stem end lengthwise into 1-1/2-inch-thick wedges, and cut the wedges crosswise into 1-1/2-inch pieces (the leaves will separate). Cut the leafy half in the same way. Keep the leaves and stems separate. Repeat with the remaining half cabbage. You should have about 5 cups stems and 4 cups leaves.
Heat a wok or a 12-inch skillet over high heat for about 45 seconds and then add the oil, swirling it to coat the pan. When the oil is hot and shimmering, add the garlic and the white stems. Stir-fry until the stems brown lightly in spots and begin to release some liquid, about 2 minutes. Add the fish sauce, sugar, salt, and chile and toss. Continue to stir-fry until the stems are barely tender, about 2-1/2 minutes.
Add the cabbage leaves, stirring quickly to move them to the bottom of the pan. As soon as the stems are just tender and the tops are barely wilted, 30 to 40 seconds more, remove from the heat and stir in the basil and 2 tsp. lime juice. Season to taste with salt, garnish with the scallions, and add more lime juice, if you like. Serve immediately.