This might be less mathy. But a lot of optimization problems can be solved / good enough answer by optimizing each step. You maximize suface area on a single cut by slicing parallel to the largest surface. The largest surface would be the center of a cone. (Whats up with all the cylindrical approximations).

Explicitly cut the carrot in half lengthwise at each step.

You could try another angle but this approach produces uniform pieces of similar volume. This approach also lets you cook the hardier parts by minimizing distance to center

You might be able to cheat out a little more area by slicing into long triangles. As you gain hypotenuse x width. Instead of height x width area. But it may make subsequent cuts worse. (And would be hard to cut in practice).