It is possible... but these companies spend billions upon billions of dollars to perfect this technology. If they found out you could do it for a lot cheaper, they'd send a hitman to your house with lots of candy to bribe you
The basic concept is this:
Processors are built in layers of silicon wafers.
Each layer is heated and treated with a boron gas, which cultivates a silicon dioxide(semiconductor insulator) and is then treated with a layer of photoresist, which becomes soluble in an ultraviolet light.
This ultraviolet light is used in a process called photolithography, where an imprint of a mask(a stencil of semiconductor architecture) can be transposed onto the silicon... similar to a photocopy. The photoresist that isn't under the mask gets exposed to the light, and turns into a liquid substance, which is then etched off with chemicals.
After that, the rest of the photoresist is removed, leaving trace patterns(ridges) of silicon dioxide.
A second layer is added by creating a thinner layer of silicon dioxide is grown over the trace patterns and etched areas of the wafer.
Then a layer of polysilicon(conductive silicon) and a layer of photoresist are applied. That layer is then treated under ultraviolet light, with a second mask, exposing a new pattern.
The photoresist is disolved, exposing polysilicon and silicon dioxide, which are etched away... leaving a trace pattern of conductive polysilicon and silicon dioxide(which again, acts as an insulator).
Ion implantation is then used to dope areas of the silicon wafer with metal ions, causing the areas to conduct electricity.
The layer process is repeated, creating windows that allow each layer to connect... and then metals such as copper and aluminum are used to fill the windows and connect the layers.
This is only how to make a basic microprocessor. Architecture design is really not explained by either AMD or Intel, although I'm sure there are plenty of sources online that will tear down even the latest and greatest processors to explain what is actually going on in them.
If you find such resources, let me know, because I'm just as curious as you.