Knee conundrum

2 things spring to mind: Crank length. If you have longer cranks on the road bike you are getting a more acute angle in the knee before appliying power. This will stress the vastus medialis (the muscle you're talking about) more. However, I think it's likely to be the more bent over position on the road compared to your MTB and (likely CX) that is causing the major difference. Go for a shorter stem and slightly more upright position more akin to your CX bike than a normal roadie and see if that helps.

Incidentally, DOMS is simply muscle soreness the day (or 2) after a ride, that's all. Secondly it would be extraordinarily unlikely for nutrition or hydration to be the issue.

As per above, you mentioning moving the pedals over, are you using the same shoes and if so, is the seat to pedal measurement the same. Another factor may be nutrition, if you're working much harder then you may be getting fatigue or dehydrating etc. and the pedalling then causing damage... 

That's fairly classic, when I had a bike fit it made things worse!

Read up on bike fits yourself, especially cleat positions and float.  Then find a 5 mile loop around where you live and keep riding it while making tiny adjustments to your cleat positions (after each loop) until the pain goes away.

For me, cleats further forward and an ever-so-tiny toe-in on the left sorted my knee pain.

It's useful to look how your knees are tracking while you pedal, especially under load or when fatigued.

Road riding is harder - much harder - on things like that because it's much more consistent same position grinding effort all of the time.  If you are enjoying yourself and doing it properly that is.  Assuming your fit is competent, I'd start by doing a ride and making sure your cadence is 10+ above how you normally ride, and see if that makes any difference.  If so, chances are you need to do some specific exercises to balance up your muscles power - see physio for details.

Echo most of the above.. I've had cleat fitting and bike fittings, and it's sorted me out (mostly).

I wouldn't go wih just one fit but I'd see an expert and explain your issues.  My first fit was in a fitting jig and it never picked up a collapsing arches (causing knee pain) or a discrepancy in leg length (causing hip pain).  A physio picked up the collapsing arhes and a subsequent bike fit identified the very slight discrepancy in leg length.  Incidentally I've had another fit which has lowered my saddle a bit and that seems to be perfect for my joints and when I tried a power meter that messed with my q factor I found no issues there and finally bought new shoes (perhaps changing my cleat allignment).  The lower position seems to put leass stress on my knees but that might be just me.

You say you have had a bike fit. Was this a simple measure you up to the bike (angles/ length etc) or was this an in depth analysis of your biomechanical state too?

I ask as a top level bike fit will analyse your flexability, strength and motor skills as well as assessing you for historical and current injuries. Knees are one of the most complex joints in the body and can be affected by ankles and hips primarily.

It may be worth seeing a physio in order to have a full body health check and put your mind at ease.

Ideally you would want to use the same seat height to the bottom of crank position on both bikes also crank length the same. Personally I do have different crank length across two bikes and it doesn't bother me Also check seat forward or backward position. Then you could measure the pedal spacing across the crank.

If you've got a friend with a Gopro, get them to film you from behind to see if your legs and knees track correctly. I friend of mine after I recorded a ride out spotted he had quite severe leg track issue, which he wasn't aware of until he saw himself riding. Items like shims may help or physio to strengthen certain muscle groups to correct an issue