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Remove the broccoli florets from the larger stems and cut into smaller pieces. Remove the woodiest part of the stem — usually ½-inch from the bottom — and discard (or save for stock). Using a vegetable peeler, peel away the thick tough outer layer of the stems, and then grate or slice into small matchsticks. Place the broccoli in a very large bowl (you will have 6 to 8 cups).
Add the sliced green onions, parsley, raisins, and sunflower seeds to the broccoli and toss to mix.
In a separate bowl, stir the mayonnaise, vinegar, mustard, salt, and pepper together. Taste for acidity and seasoning then adjust as desired. For a little extra flavor, add two to three small dashes of toasted sesame oil (see notes). Pour two-thirds of the dressing over the slaw, and then mix well.
If the broccoli slaw seems dry, add a little more of the dressing. Eat right away or let it sit in the refrigerator for about an hour to let the flavors mingle.