What are future steps that can be taken in order to learn more about the inner workings of a Neutron Star such as "nuclear pasta" and the materials in the inner core?
An incomplete shortlist would be:
(i) If you can measure the maximum possible mass of a neutron star (the most massive neutron star gives a lower limit to that), then it tells you how "hard" or "soft" the equation of state (the relationship between pressure, density and composition) is in the interior (especially at the core), and therefore tells you something about the matter composition and the nature of the strong nuclear force at small separations. Nuclear pasta is unlikely to have a dramatic effect on this.
(ii) Measuring the masses and radii of neutrons stars would directly tell you about the equation of state at different depths inside a neutron star. Again, I think nuclear pasta does not have a major bearing on these diagnostics.
(iii) Measuring the rate at which neutron stars cool, tells you about the interior composition. Quark matter and boson condensations in the core would allow for faster cooling. Abrupt changes to the cooling rate can tell you about superfluid phase transitions. Cooling can also be accelerated by nuclear pasta phases by allowing the URCA process to operate.
(iv) Measuring the response of pulsars to glitches tells you about superfluidity in the interior and crust and the coupling of crust and core will depend on the properties of what is in between - i.e. nuclear pasta.