Conception and manufacture of the Avogadro Project's (nearly) perfect silicon sphere (and, what is “atomic feeling”?)
Short answer:  The success of the project benefitted from many contributors.

According to BIPM and its Avogadro project, there are two spheres (reference).  The page equally refers to a publiction about the 2011 $\ce{^{28}Si}$ atom count (paywall) and the report in 2017 (here, open access).  From the later paper, you may identify the origin of the two spheres in the Czochralski crystal obtained initially:



as well as where samples were taken to determine the amount of carbon, oxygen, nitrogen, and boron (loc. cit., table 1), the molar mass of silicon (table 2), concentrations of point defects in the lattice (table 3).  Equally, the analysis took into consideration the $\ce{SiO2}$ layer naturally formed if the sample is exposed to air, as well as adsorbed water; for both, the upper limit of layer width as well as the mass are provided.  As an example, the deposit of oxygen on the spheres was mapped by X-ray photon spectroscopy (XPS):



The article describes that the diameter of the spheres was assessed by a sphere interferometer



in vacuum at $20\,^\circ{}\mathrm{C}$ within a tolerance of $\pm 3\,\pu{mK}$ to allow the statement


  The measured diameter was the apparent diameter, which is not corrected for the phase shift due to the surface layers. The mean apparent diameter was 93 723.723 61(61) μm.


As for imperfections, this property equally was mapped:



With papers titled like The self-weight deformation of an x-ray interferometer the dedicated issue of Metrologia 48(2), April 2011 shed some light on monitoring the quality of the samples and the used tools for the analyses.

Additions:
A video of channel veritassium is dedicated to the reference spheres, too.  The section starting at 7:00 min focusses on carving and polishing them (8:11 min, at 8:19 min likely showing Leistner himself).

A now open-access publication by Fujii et al. equally briefly describes the stages to carve-out the spheres:



Starting from the ingot, a section is cut.  A lathe is used to obtain a sphere already with an shape accuracy of $1\,\pu{mm}$.  To smoothen the surface, these spheres are lapped with alumina particles in water with finer and finer grains; eventually (fine lapping) within a margin of $100\,\pu{nm}$.  To polish the surface, collodial alumina and eventually titanium dioxide yielded a shape deviation of less than $50\,\pu{nm}$ and a surface roughness below $0.2\,\pu{nm}$.  This publication explicitly refers to Leistner's work including publications authored by him (example, example) and then established protocols easing to manufacture such samples reliably.

PTB continues both further development of the interferometers (here) and the deployment of these spheres as substitute of the old standards (example).  (As of writing, the mass standard's group seeks collaborators for Si spheres (reference, perma).)  Comments by Maurice and GM suggest the advancement of mechanical engineering and this new definition of the kilogram might offer smaller mass references which do not need to be as large and heavy as $1\,\pu{kg}$.