"Just" two tests...
The OP would like to have an independent FAA Private Pilot Certificate. (Certificate, please, not license.) I won't ask why he wants to do that. I won't suggest alternatives.
• Passing a knowledge test. The test requires an instructor's endorsement.
• Meet certain aeronautical experience minimums and pass a checkride.
Assuming all specific experience requirements have been met (Part 61.109), the OP must fly three hours in preparation for a checkride within the previous 60 days. A couple of specific requirements someone from Europe might not have yet met would be three hours of night flight and three hours of simulated instrument flight. (These can overlap with each other and be part of the three hours prep. Can be done on a cross country flight, too.)
I don't believe that European flight training includes what the FAA-world calls "ground reference maneuvers." The three hours of checkride prep should easily cover this for most current pilots.
The FAA's Practical Test Standards have been replaced by Airman Certification Standards (ACS) at the private pilot level. Read the ACS to know exactly what the examiner is to test you on. If you know the material and can fly to ACS standards you should pass the checkride.
The FAA accepts all flight time. If it's documented, it counts. I've seen European clubs where pilots keep their own logbook, but training is recorded elsewhere - w/o an instructor's signature in the logbook as the requirement in the US. Probably not a problem.
Where do you plan to do this? It can be done in Europe with effort. It can be done in a non-N-reg airplane. It can be done at a decent flight school in the US in just a few days. I'd budget four or five days and expect 1) more ground instruction than flight time and 2) maybe a couple hours flight time more than the required three to be sure.
A good instructor can make each of three or four flights a practice checkride and train on weak areas. A good instructor will also try to fulfill more than one requirement at a time where possible - such as a night cross country under the hood.
If you plan to use this FAA Private Pilot Certificate outside the United States you will also need a Restricted Radiotelephone Operators License.
For flights within the US in the very near future, no medical will be required. Flights outside the US will still require a medical.
What have you been flying? If it's one of the "standard" Cessnas or PA-28 family, it should be an easy transition.
Hope this helps.
Terry