For the dust: Place a rack in the middle of the oven and preheat the oven to 250 degrees F. Pulse the dill and salt together in a food processor until the dill is finely ground and combined with the salt. Spread the dust in an even layer on a sheet pan and bake until dry, about 30 minutes.
For the tots: Place the potato in a medium pot of cold, salted water and bring it to a boil over medium-high heat until a knife inserts easily into the potato but the interior is still firm, 5 minutes. Transfer the potato into a colander and let cool slightly. Peel and coarsely chop the potato. Then process the potato in a food processor to very small chunks, 10 to 15 pulses. In a large bowl combine the flour, 1 teaspoon salt and the pepper. Add to the processed potatoes, mixing well with a spoon or spatula to distribute the flour mixture evenly through the potatoes. Form spoonfuls of the mixture into 2-inch logs.
Line a baking sheet with paper towels. Heat the oil in a medium saucepan over medium-high heat to 350 degrees F. Fry the tots in small batches until they are golden brown and crispy, 2 minutes per side. Remove the tots from the oil with a slotted spoon and transfer to the baking sheet. Sprinkle the still warm tots with the dill dust. Continue frying and dusting the remaining tots.