Peel the potatoes, and grate them on the coarsest side of a grater. In a large bowl, cover them with cold water; drain, and repeat until the water runs clear. Dry the potatoes thoroughly, using a clean dishtowel to wring out as much excess as you can.
Put the oil in a large cast iron or nonstick skillet over medium-high heat. When the oil is hot, add the chorizo and use your spoon to break it apart into an even layer. Cook for 15 to 18 minutes, stirring occasionally but not too often, until it’s deeply browned and most of its fat has rendered. With a slotted spoon, strain out and reserve the sausage, leaving the oil behind.
Add the potatoes to the skillet, lightly toss to coat with the oil, then spread them in an even layer and leave them alone for 5 minutes or so (they won’t crisp up if you keep stirring). When the bottom is crisp and golden, give them a stir and keep frying until they're broken up with nicely browned edges. Remove from the heat, and combine with the chorizo.
Pit the avocado, and cut it in a small crosshatched pattern. Coarsely mash it with the lime juice, then fold in the onion. Set aside.
Just before serving, toast the tortillas directly over burners on medium-high heat until the edges are slightly charred, 1 to 2 minutes per side. Spread a spoonful of mashed avocado over each tortilla, fill with the chorizo and potatoes, and top with radishes.