I essentially want to wash away the excess water and already consumed flour after I've fed my culture. What I usually do is whisk the culture and pour off the excess down to 1 cup. I then add more flour and water and then store it in a mason jar in my refrigerator until I feel like baking again. This works, but I've always been curious how the commercial factories create a yeast cake that is sold in some grocery stores or specialty shops.
Food Grade Yeasts, Food Technol. Biotechnol. 44 (3) 407–415 (2006) describes the process by which yeast is commercially prepared in good detail for an overview.
There are several stages of cultivation and growing and increasing the yeast biomass, culminating as its final steps:
Treatments and packaging The yeast in the final trade bioreactor is concentrated by centrifugation and finally harvested by a filter press or a rotary vacuum filter, until it contains 27–33 % of dry cell mass. The yeast cake is blended with suitable amounts of water and emulsifiers and cutting oils (soybean or cottonseed oil) to obtain its extrudable form. The yeast is then packaged and shipped as compressed fresh baker’s yeast, or thermolysed and dried to form various types of dry yeast. The dried yeast is packed under vacuum or nitrogen atmosphere. The packaging method varies among manufacturers and depends on the type of yeast product.
Note that this process requires assorted items of industrial or laboratory equipment which may not be suitable to the home environment, including centrifuges and filter presses or vacuuum filters.
If you really only want to preserve your starter strain without maintaining an active starter, simply drying a sample and then storing it in the freezer may be a better option. This process is described at Breadtopia.
TL;DR: You can't.
The primary reason why this won't work is that commercial yeast and sourdough cultures are totally different species. Commercially available yeast is almost always Saccharomyces cerevisiae whereas a sourdough culture will be a symbiotic mixture of one or more acid producing bacteria and one or more yeasts (usually a Lactobacillus along with a Candida and/or Saccharomyces yeast).
Cake yeast is a commercially manufactured product consisting of little more than yeast, with a bit of carbohydrates, and some binding agents. Your starter is a mix of a lot of carbohydrates, a lot of water, and some microorganisms. Getting it to the same concentration of microorganisms without some serious lab equipment would be pretty much impossible. Even if you did accomplish that, you couldn't use your sour cake the same as compressed yeast as the microorganisms are too fundamentally different (you'd have to reconstitute the sour yeast and give it at least one feeding before using it).