Lucas, The white smoke is combustion gases that have leaked past the piston rings,when the engines fires the fuel mixture in the cylinder. A small amount of leakage occurs even in a healthy engine. The reed valve (#3) allows the blowby to be expelled from the crankcase,but prevents a reverse airflow into the crankcase. The result being that a Kohler (and many other engines) run with a slight negative (less than atmospheric ) pressure in the crankcase. See the Kohler engine manual,section 4 pages 2&3 on how to measure for a negative. A slight "pulsing" at the cover might be found,as long as there is a "negative" in the crankcase. If you follow the test procedure and know that the breather assembly is installed correctly, and get a positive reading, worn rings,cracked piston or scuffed cylinder walls are likely causes. Also make sure the crankcase is not over full of oil.
The breather assembly must have all the parts and be assembled correctly to work. Part #2 has a small drain back hole that must be at the bottom. Part #6,the filter, uses the coaslescing principal to remove the oil vapor from the gases and allow it to return to the crankcase through the drain back hole in #2. Part #3, the reed, must lay flat and seal the two large holes in #2,when the parts are at rest. Let us know what you find.