For the pancakes: Combine 4 cups cold water and 1 tablespoon kosher salt in a large bowl and stir until the salt dissolves. Stir in the sweet potato and let sit for 15 minutes. Drain, then dump onto a large piece of cheesecloth and wring out as much water as possible.
Preheat the oven to 375 degrees F. Whisk together the eggs, onion and some pepper in a large bowl. Add the sweet potato and flour, and mix until combined. Fold in the green onions.
Heat a few tablespoons each of butter and oil until shimmering in a large nonstick skillet over medium heat. Using a scant 1/4 cup per pancake, add the sweet potato mixture to the pan, leaving some space in between. Flatten slightly and cook until golden brown on both sides, about 3 minutes per side. Drain on paper towels, then transfer the pancakes to a baking sheet and place in the oven for 5 minutes to finish cooking.
For the eggs: Fill a high-sided skillet halfway with water and bring to a boil over medium-high heat; add the vinegar. Line a plate with paper towels. Break each egg into a little ramekin or cup.
One at a time, slip the eggs into the water, leaving space between them; adjust the heat to a bare simmer. Cook until the whites are set and the yolks still runny, about 4 minutes. Using a slotted spoon, transfer the eggs to paper towels to blot off the water.
To serve: Put the pancakes on a serving platter. Top 4 of them with a smear of pepper jelly, a few slices of ham and a poached egg. Season with salt and pepper. Top with another pancake to make sandwiches (see Cook's Note).
Alternatively, you can serve the pancakes open-faced, with extra pepper jelly. Double the number of poached eggs if you do.