Turn your grill on high heat and let it heat up for at least 10 minutes. Once it’s heated up, reduce the heat down to medium-high.
In a large pot, add the potatoes, enough water to cover them by 2 inches and generously salt the water (like pasta!). Bring to a boil, then reduce the heat to medium-low and simmer for 10-20 minutes (depending on size), or until easily pierced with a knife. Transfer the potatoes to a large cutting board to cool for a bit.
If not grilling the potatoes, then just cut them into bite sized pieces. To grill the potatoes, cut them in half and brush a bit of olive oil on the cut-side just to lightly coat. Place them cut-side -down on the grill at the same time as the cabbage and scallions (see below).
Add cabbage wedges and scallions to a large bowl and toss them in 1 tablespoon oil and ¼ teaspoon salt. Add them to the grill, then cover and grill for about 5 minutes. The scallions should be softened and charred in spots and the potatoes should have nice grill marks. Transfer those to a cutting board. Flip the cabbage and grill for another 2-3 minutes, covered, or until charred on the other side, then transfer to the cutting board.
While the veggies cool a bit, in that same large bowl, whisk together the remaining ⅓ cup oil, vinegar, mustard, and remaining ¾ teaspoon salt. Cut the potatoes into bite-sized pieces and transfer to the bowl and shake them around to coat. Thinly slice the scallions, removing the hairy root ends, and chop up the cabbage (remove the core before chopping). Transfer to the bowl along with the pickled jalapenos and toss to coat. Taste the salad - if it tastes flat, add another tablespoon of vinegar and/or more salt.
Serve it up topped with some black pepper.