Mix masa harina, 1 1/2 cups warm water and masa salt into a soft dough. Knead for 2 minutes, adding reserved water a teaspoon at a time only if the dough cracks when pressed, then cover and rest for 15 minutes; resting is included in prep time.
Heat picadillo oil in a skillet over medium-high heat. Brown beef and onion for 8 minutes until beef reaches 160°F. Drain excess fat, then stir in potato, carrot and garlic for 1 minute.
Add tomato sauce, broth, cumin, oregano and picadillo salt. Cover partly and simmer for 15 to 18 minutes until potato and carrot are tender, then uncover for 2 minutes if needed so the topping is thick enough to spoon.
Meanwhile, divide masa dough into 12 balls. Flatten each between sheets of plastic into a round about 3 inches wide and 1/3 inch thick, keeping unshaped dough covered.
Heat a dry griddle over medium heat and cook rounds in 3 batches for about 2 minutes per side until their surfaces set with light brown spots. While each is still warm, use a clean towel to protect your fingers and pinch a raised rim around its top edge.
Heat shallow-frying oil in a large skillet to 350°F. Fry shaped bases in 3 batches for 1 to 2 minutes per side until lightly golden and cooked through, then drain on a rack.
Divide warm picadillo among the 12 sopes, add a little shredded lettuce and divide crema evenly over them. Serve 3 sopes per person immediately.