I have a Cape Cod house in the Northeast that I have just dormered out the
Second floor with cathedral ceilings. I have an oil forced air
Furnace/central air and there are currently two supply and two return ducts
Going to the second floor. There is one zone in the entire house that
Controls both the first and second floor.

The problem is that the heating is now inadequate on the second floor with
The dormer/cathedral ceilings. When we turn the thermostat up, the first
Floor gets very hot and the second floor does not warm up enough.

I was prepared for the AC not to be adequate, but the heat I thought would
Be okay. I'm looking to solve the problem in the least expensive and
Energy efficient way. I have just been reading about hydro air system,
However unclear if that would be appropriate? I have also heard of an
Outside heating element that runs up the side of the house. I am looking
To explore options without having to create separate zones on the current
System.

Look into mini split units. They can be used for both heat and cool. Small in size but big in performance they fit in with your problem for a easy solution.

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Lets look at this in two steps. The first at getting heat to the second floor and the second being a fix.

You can increase your blower speed and damper down the supplies to the first floor. You can run your fan continually to help stratify the air temperature or you can install duct boosters to the second floor. This is only a temporary fix.

When you added the dormers, you added a lot of outside wall space that is exposed to the cold and hot temperatures. Hind sight being 20/20, I guess you should have consulted a heating contractor prior to the expansion. Now your heating system is compromised, the cost to condition your home will double and you will either need to upgrade it and the ductwork or do something like hvac1000 suggests, add a separate system for the second floor.

Hydro air is not the answer, you will need to add a boiler and an air handler in a space that you already use for living space. I recommend the ductless avenue as what was suggested. It's not cheap, but at this point any money spend to solve the problem will have to go as a "learning experience"