Bring a small pot of water to a boil, over medium heat and add the potatoes. Season the water with salt, to taste, and cook until a knife inserted into the center of a potato meets some resistance, about 5 minutes. Drain well and put the potatoes onto a baking sheet lined with paper towels.
Heat the 2 tablespoons of unsalted butter and canola oil in a medium cast iron pan over medium-high heat until the butter begins to sizzle. Add the par-boiled potatoes, season them with salt and black pepper, to taste, and the smoked paprika and fry until they are golden brown. Line the same baking sheet with clean paper towels and transfer the potatoes from the skillet with a slotted spoon to the sheet.
Melt some additional butter to a 6-inch nonstick saute pan over medium heat. Season the beaten eggs with salt and black pepper, to taste, the fresh parsley and thyme leaves. Add 1/4 of the mixture, about 3 eggs, to the pan, and lightly scramble them using a fork. Spoon a line, along the center of the omelet with some of the piquillo peppers, Manchego cheese, soft goat cheese, if using, and the potatoes. Roll the finished omelet onto a serving platter and garnish it with some parsley and thyme sprigs.