Pour 8 cups water into a large saucepan and heat over medium-high until it reaches a rolling boil.

While the water heats, peel the 3 large Russet potatoes and cut them into 1-inch cubes.

When the water is boiling, add 1/2 teaspoon baking soda, then add the potato cubes. Return to a boil and cook for 1 minute.

Immediately drain the potatoes in a colander. Return the drained potatoes to the empty saucepan and set the pan over medium heat. Stir or shake the pot until any surface moisture has evaporated and the potatoes are dry, about 1 minute—do not brown them.

Spread the potatoes in a single layer on a rimmed baking sheet and let them cool and dry for a few minutes (about 3–5 minutes) until they are no longer steaming.

Meanwhile, heat 4 cups peanut oil or vegetable oil in a large, heavy pot to 375°F (use a thermometer). Do not overfill the pot; leave room for the oil to bubble when potatoes are added.

Fry the potatoes in batches: add about one handful of potatoes at a time to the hot oil (be careful of splashes). Stir occasionally with a wire skimmer or large slotted spoon to keep pieces separate. Cook each batch 20–25 minutes, until the potatoes are golden brown and crisp. Allow oil to return to 375°F between batches.

Remove fried potatoes with a skimmer or slotted spoon and transfer to a paper towel–lined baking sheet to drain. Immediately sprinkle with Salt to finish to taste.

To make the sauce, combine 1 cup mayonnaise, 2 teaspoons minced garlic, 1 teaspoon smoked paprika, 1/2 teaspoon sea salt, 1/2 teaspoon cayenne pepper, 1/4 cup tomato paste, 2 teaspoons sherry vinegar, and 1 teaspoon lemon juice in a bowl. Whisk until smooth and well combined.

Serve the hot fried potatoes with the sauce on the side.