You are probably referring to the Feynman-style double-slit experiment, done with individual electrons. A recent experimental realization of this was reported in
Controlled double-slit electron diffraction. R Bach et al. N. J. Phys. 15 033018 (2013) (open access).
and it is described in more readable detail in
Feynman's double-slit experiment gets a makeover. H Johnston, Physics World, 14 March 2013.
The experimental data looks like this, which you might recognize:
Image source: Bach et al., CC-BY license.
The experiment is in essence really simple: simply take an electron gun (i.e. a cathode-ray tube like you find in old TVs, simply a heated piece of metal that's negatively charged) and dial it down to bring the current to levels where there is only one electron in the system at a time. If your detector is sensitive enough, then you will be able to detect individual electron hits.
The detector in this experiment was a microchannel plate (a metal plate with lots of tiny holes in it), to discretize the electron positions, with a phosphorescent screen behind it to change the electron hits into light, which is then imaged with a (fancy, low-light, but otherwise perfectly normal) camera.
Note, however, that this electron-counting technique only tells you "here's one electron" and "there's another one" and so on. Counting the number of electrons in a given charged object is a lot more difficult. If you want to do that then you can probably achieve single-electron accuracy in objects with tens or possibly hundreds of electrons, depending on the situation, but beyond that you really need to ask exactly what it is you're doing and why.
In particular, it is neither feasible nor reasonable to put together one Coulomb's worth of electrons to single-electron accuracy.
This is partly because, in the end, measurements of charge are not that useful - it is measurements of current that really matter. It is possible, though, to construct experiments that will measure current by observing single electrons pass in single file through a series of Josephson junctions, like the one in
Current measurement by real-time counting of single electrons. J Bylander et al. Nature 434, 361 (2005), arXiv:cond-mat/0411420.
The authors claim to be able to measure currents as large as 1 pA by counting single electrons, which corresponds to 6 million electrons per second.
The current state of the art is that these experiments are approaching the same level of accuracy, precision, stability, reliability, and dynamic range that one can get with traditional current calibration (which is based on the force between two parallel wires, and therefore depends on the SI standards for time, length and mass).
This means that we can turn the relation around, and user experiments like Bylander's to define the ampere as the current caused by the passage of 6.241509×1018 elementary charges per second. (This now depends on the SI second but nothing else.) And indeed, the plan is to do just that, as part of the "new SI" as proposed by the BIPM. This is part of a larger makeover of the SI (including a redefinition of the kilogram in terms of a set value for $\hbar$ by means of watt balances, and re-workings of the kelvin and the mole), so it's coming along but slowly. (In particular, it is not quite clear what the proposed realizations actually are, but it will happen.)