Why are my doughnuts raw in the middle? I recently began making doughnuts. I did a few experiments.

The first time I followed the recipe exactly and they came out perfectly. Large, airy, and soft. But on day 2 they were too dry and lost some of their texture.

To address this problem I did 2 half batches as experiments. One I added too much butter and the other I used my sourdough starter instead of yeast.

None of the butter batch cooked properly. All had raw centres. Only 1/4 of the sourdough batch came out right. The rest were also raw in the centre.

I purposely weighed smaller amounts for my test batches. They were 10-15gr lighter than my first batch. They were also less airy. I fried them the same as far as I can tell. 

Why are the smaller doughnuts coming out raw?
The answer is simple: your test batches were less airy, and more dense.  The heat from the oil was likely unable to penetrate the dough as effectively and cook the interior.

If the recipe turned out properly the first time around, that's probably not the issue.  If your concern is with storing the doughnuts  (though how you would have any left is a mystery to me!) then you may want to check out more effective methods of storage.  There may be some helpful guidelines in this thread.