Preheat a smoker to 250 degrees F. Preheat the oven to 375 degrees F.
For the Brussels sprouts: Toss the Brussels sprouts, olive oil, 2 tablespoons salt and 1 tablespoon pepper in a large mixing bowl and then spread out on a baking sheet. Roast until lightly charred and tender, 15 minutes. Remove from oven and toss in the sherry vinegar while warm.
For the smoked sweet potatoes: Toss the sweet potatoes with the olive oil, brown sugar, red pepper flakes, 2 tablespoons salt and 1 tablespoon pepper in a large mixing bowl. Spread evenly on a rack over a baking sheet and smoke for 20 minutes (see Cook's Note). Finish cooking in the oven until fork tender but still with a little bite, 10 minutes.
For the vinaigrette: In a small bowl, whisk together the olive oil, sherry vinegar, agave and mustard in a small bowl; season with salt and pepper.
To assemble the salad: In a large bowl, toss the Brussels sprouts with the sweet potatoes. Pour in the vinaigrette and toss well. Serve over a bed of arugula, shower with pecorino and scatter with Pickled Red Onions.
In a medium sauce pan over medium-high heat, combine the beets, red wine vinegar, brown sugar, salt, peppercorns, bay leaf and 1/2 cup water. Bring to a boil and cook for 7 to 8 minutes. Place the red onions in a large mixing bowl and pour over the hot liquid. Allow to cool, then transfer to an airtight container and refrigerate. Remove the beets before serving. Keeps for 2 weeks in the refrigerator.
If you don't have a smoker, create a foil pouch with 2 cups soaked and drained wood chips. Place on the stovetop over high heat until smoking. Carefully place the pouch in a large cast-iron skillet, cover with a cooling rack, and pile cut and seasoned sweet potatoes in the center of the rack. Place a second, inverted cast-iron skillet over the potatoes to create a lid. Smoke for 20 minutes, then transfer to a 375-degree oven to finish cooking, another 10 minutes.