R."We increased throughput 300 pieces per hour. As a result, we can now process 4200 pieces during a 4 hour span."
This is good English. There is nothing wrong with it.
I mean the "As a result" part. (It's not clear whether you mean you increased it by 300 pieces or to 300 pieces. It might be clear where you come from. I guess it's 'by'. Also, '4,200' and 'four-hour span' would seem more standard to me.)
Is there a rule being broken here? No, not in the structure you're referring to.
The readability can be improved by simply combining the sentences:
I disagree. The first version seems more readable to me. Two short sentences are usually more readable than one long one - as long as they are meaningful, which this pair is.
"As a result of increasing throughput 300 pieces per hour, we can now process 4200 pieces during a 4 hour span."