Let’s say we have an unlimited budget and we don’t care much about practicality, and our goal is to be able to receive an egg, which has been boiled for an unknown duration, and determine its doneness, which I would define as how liquid vs solid the yolk is, without damaging the egg or rendering it inedible.
what are are options for building a device or otherwise making this determination?
Spin the egg then ever so briefly put your hand on it to stop it. If its boiled (solid) it will stay stopped. If its raw (liquid) it will resume spinning. But as you mentioned, it can’t tell the degree of solidness or liquidness.
As long as practicality is not an issue, just cut the egg in half and see how done it is. If it is not done to the desired level, discard it and do the same with the next egg until one is found that is cooked the right amount.
Optically transparent and X-ray transparent are not related. X-radiography is sensitive to density, which is probably not much different between raw and cooked. Small-angle X-ray scattering is sensitive to structural changes, but in order not to waste eggs, you’ll want to use the full USAXS/SAXS/WAXS suite at Argonne National Lab.
Whilst this wouldn’t work by hand for partly cooked eggs (pretty sure it’s the white that’s doing most of the work with this trick) - the effect might be so slight for a soft-boiled egg that it would not resume spinning, but a machine that precisely senses the momentum of the contents of an egg by rotating it and stopping could probably be made.
Attach an audio transducer to side of egg. Tap on other side of egg (without breaking shell) using a calibrated striker. Measure response at the transducer - possibly take an FFT or just measure rise and fall times. Frequency response will be a function of viscosity. Compare to empirical data taken from eggs of various degrees of doneness.
Place 6 eggs and 6 egg analogues into a pot at the same time (carefully ensuring that the egg and egg analogues are stored together so they start from the same environmental conditions). Starting at 5 minutes, retrieve 1 egg and 1 egg analogue every 30 seconds. Destructively test the egg analogue to gather, with high confidence, what the state of the egg is. You can serve the egg for which the analogue is the closest to your desired doneness.
I believe we can bioengineer an egg analogue to have almost indistinguishable behavioral characteristics to a real egg and, with sufficient research, could even get the price down to the cost almost exactly at parity to real eggs. If we were to ever achieve this feat, we could package 6 eggs and 6 egg analogues into cardboard dimpled containers and sell them at the same price as a dozen eggs!
We could construct an egg analogue containing a material that matches the thermal coefficient and conductivity of an egg, and which contains a material that goes transparent at the temperature at which egg protein starts to coagulate. Put it in a transparent shell. Drop it in the water with your real egg. You will see a surface inside the analogue that shrinks as it heats. If the surface shrinks to nothing the egg is likely hard boiled. With some of the apparent analogue contents still visible, you can see how far the transition temperature for cooking protein has travelled into the eg, and can decide on how cooked you want it. This mimics the actual heat transfer and cooking, and probably contains all the information you need. It would even account for cooking in less than boiling point water (or boiling at altitude.)
Some sort of phase transition material does the temperature transition, just needs to be reversible. Maybe a change in refractive index, then you mix a dispersion of the materials into the main matrix. When it hits the right temperature, you choose the surrounding matrix such that its refractive index matches, and the thing transitions from milky to clear.
A wax that solidifies with a messy structure and is opaque, versus transparent when melted, could also work. A lot of possibilities.
We might need two temperature transitions, one for egg white, and another for yolk.