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To my understanding the mole is the unit used to translate between mass on the atomic level and mass in the macro level, defined as the number of atoms in $12$ grams of Carbon-$12$, which apparently turns out to be $\pu{6.0221409e23}$ atoms, and this was defined as Avogadro's Number.
How was this number derived? Is it something I can derive myself? If I had $12$ grams of Carbon-$12$, how would I be able to deduce that it had $\pu{6.0221409e23}$ atoms?
To determine Avogadro's number you have to measure the same unit at the atomic and macroscopic scales.
This was first achieved by Millikan who measured the charge of an electron. The charge of one mole of electrons was already known and is a Faraday. Dividing both, you get Avogadro's number.
Before that, Josef Loschmidt was the first one to calculate the number of particles in a cubic meter of gas, using the kinetic theory of gases. This is, of course, also related to Avogadro's number. Avogadro himself did not give any number. He just stated that equal volumes of different gases at the same pressure and temperature have equal amount of particles.
A more modern way of doing it is determining the density of an ultra pure element and then determine the number of atoms and their distances in a unit cell with X-ray diffraction.
So yes, you could, in principle, derive Avogadro's number with pure monocrystalline carbon 12. You will, however, need very good instrumentation.
As per the answer to this question on Chemistry SE it might become the other way round. The Avogadro number will be absolute (defined, with 0 error) and then the kilogram will be redefined as a function of the number of atoms in a monocrystalline perfect $\ce{^{28}Si}$ sphere.
There is a school-level experiment that can estimate Avogadro's number
Avogadro's number is now defined to be a precise constant. But the question is whether simple experiments can be used to estimate it. The historic experiments designed to estimate it are not always useful as they usually involve techniques or equipment unavailable outside fully-equipped laboratories. Like the indirect method used by Millikan (measuring the charge on a single electron in the famous oil-drop experiment which, via known electrical measurements, can be converted into an estimate of the constant) or the estimates based on x-ray measurements of ultra-pure crystals which estimate density via measuring atomic distances in a known mass).
But a simple experiment can get a good approximate estimate given a small number of assumptions based on an 1890 idea from Lord Rayleigh. The experiment involves spreading a known quantity of a pure surfactant over a water surface and measuring the area it occupies. The assumption is that the surfactant occupies a compact monomolecular layer on the water surface (there are other ways to verify this). If you put in assumptions about the size of the molecule and the thickness of the layer and measurements of the mass of the surfactant that created the layer (easy to measure with some accuracy) then a decent estimate of avogadro's constant can be derived.
The process of doing the calculation is described in this Physics.SE answer (with the caution that assuming a surfactant occupies a cube is wrong).
In the 1980s this was a common school experiment. One description of that experiment is here.
So there are simple experiments that can be used to estimate the Avogadro constant. But more reliable estimates need a lot more equipment.
