Sudden shots of cold water from shower and sink faucets
If it was just the shower, I would look at the shower balancing valve.
But you say this is happening almost identically in the kitchen?  It is possible that the valves for the kitchen are not both opening at the same time.
You could appraoch this as two separate problems, even though they seem related.  I suggest these steps:

contact the manufacturer for the kitchen no-touch, and describe the problem to them.  They will walk you through the diagnosis.
When that issue is resolved then go after the shower.  What happens when running a sink faucet and the toilet is flushed?

A likely area in general is restrictions in hot water heaters, but you get this with flushing the toilet.  What kitchen faucet model, and what kind of shower control do you have?  What kind and style water heater do you have? Tank, on-demand?  Meanwhile chase down the steps above.