Combine water and baking soda in a large bowl. Add beef and mix until thoroughly combined. Set aside.

Heat 1 tablespoon of oil in a 12-inch skillet over medium heat until shimmering. Add onion and cook, stirring occasionally, until softened, 4 to 6 minutes. Add chili powder, paprika, cumin, garlic powder, and 1 teaspoon salt and cook, stirring frequently, until fragrant, about 1 minute.

Stir in tomato paste and cook until paste is rust-colored, 1 to 2 minutes.

Add beef mixture and cook, using a wooden spoon to break meat into ¼ inch pieces, until beef is no longer pink, 5 to 7 minutes.

Transfer beef mixture to bowl; stir in cheddar until cheese has melted and mixed in well. Wipe the skillet clean with paper towels.

Thoroughly brush both sides of the tortillas with 2 tablespoons of oil.

Arrange tortillas, overlapping (if doing 12), on a rimmed baking sheet in 2 rows. Bake until tortillas are warm and pliable, about 5 minutes.

Remove tortillas from oven and reduce oven temperature to 200 degrees if making 12 tacos, otherwise turn off oven.

Place 2 tablespoons of filling on 1 side of 1 tortilla. Fold and press to close the tortilla (edges will be open, but the tortilla will remain folded).

Repeat with the remaining tortillas and remaining filling. (At this point, filled tortillas can be covered and refrigerated for up to 12 hours.)

Set a wire rack in the second rimmed baking sheet and line the rack with a double layer of paper towels.

Heat the remaining ¼ cup of oil in the now-empty skillet over medium-high heat until shimmering.

Arrange 6 tacos in a skillet with open sides facing away from you. Cook, adjusting heat so oil actively sizzles and bubbles appear around edges of tacos, until tacos are crispy and deeply browned on 1 side, about 2 minutes.

Using tongs and a thin spatula, carefully flip tacos. Cook until deeply browned on the second side, about 2 minutes, adjusting heat as necessary.

Transfer shallow-fried tacos to a paper towel-lined rack and blot tops with paper towels.

To serve, open tacos like a book and add toppings.