Place the whole scrubbed sweet potatoes in a large pot and cover with cold water. Add 2 tablespoons kosher salt.
Bring to a gentle boil, then cook until the potatoes are tender but still fairly firm, about 10 to 15 minutes — you should be able to pierce them with a fork somewhat easily but they should not be falling apart.
Drain the potatoes and let them cool just enough to handle. Meanwhile, heat the grill to medium (about 400°F).
Slice each potato in half lengthwise, keeping the skins on. Slice each half into 4 to 6 wedges (depending on potato size), aiming for wedges about 1/3 inch thick at the thinnest edge.
Place the wedges in a large bowl. Drizzle with 2 tablespoons extra-virgin olive oil, then add 1 tablespoon brown sugar, 1 teaspoon ground chipotle chile pepper (if you choose cayenne instead, use 1/4 teaspoon), and the remaining 1 teaspoon kosher salt. Toss well to coat each wedge evenly.
Arrange the wedges in a single layer on the preheated grill (work in batches if needed to avoid crowding). Grill until they develop light golden color and grill marks and are tender, flipping once with tongs — about 6 minutes total (adjust a minute or two per side as needed for your grill).
Transfer the grilled sweet potato fries to a serving platter. Sprinkle with chopped fresh cilantro if desired and serve hot.