Note that Eagle Palms are NOT considered “Ranks” by BSA.

After successfully completing your Eagle Scout board of review and being validated as an Eagle Scout by the National Service Center, you will be entitled to receive an Eagle Palm for each additional 5 merit badges you have completed before your Eagle Scout board of review. For these Palms only, it will not be necessary for you to complete any of the requirements stated below.

Effective August 1, 2017, you will receive all palms based on the number of merit badges you have earned prior to making Eagle along with your Eagle rank.
This means for example you have earned 32 merit badges since joining scouts, you have completed the 21 required for Eagle, when you earn your eagle you will receive a gold palm along with your eagle, and you have one additional merit badge earned toward you Silver palm. 

After becoming an Eagle Scout, you may earn Palms by completing the following requirements:

1. Be active in the Boy Scouts of America for at least three months after becoming an Eagle Scout or after the last Palm was earned. **
2. Since earning the Eagle Scout rank or your last Eagle Palm, demonstrate Scout spirit by living the Scout Oath and Scout Law. Tell how you have done your duty to God and how you have lived the Scout Oath and Scout Law in your everyday life.
3. Continue to set a satisfactory example of accepting responsibility or demonstrating leadership ability.
4. Earn five additional merit badges beyond those required for Eagle or last Palm.***.
5. While an Eagle Scout participate in a Scoutmaster conference.*
6. Effective August 1, 2017, no Board of Review is required to earn Eagle Palms

Notes:

*For Varsity Scouts working on Boy Scout requirements, replace “troop” with “team” and “Scoutmaster” with “Varsity Scout Coach.”
For Venturers working on Boy Scout requirements, replace “troop” with “crew” and “Scoutmaster” with “Crew Advisor.”
For Sea Scouts working on Boy Scout requirements, replace “troop” with “ship” and “Scoutmaster” with “Skipper.”

** Eagle Palms must be earned in sequence, and the three-month tenure requirement must be observed for each Palm.

***Merit Badges earned any time since becoming a Boy Scout may be used to meet this requirement.

Eagle Palms are worn by youth members on the Eagle Scout rank patch, or on the ribbon of the Eagle Scout Medal.  You may wear only the proper combination of Palms for the number of merit badges you earned beyond the rank of Eagle. The Bronze Palm represents 5 merit badges, the Gold Palm 10, and the Silver Palm 15.

Source: Bryan on Scouting – The way Scouts earn Eagle Palms is about to change

completed before a candidate’s 18th birthday. The Palms beyond Silver can’t be entered into BSA’s database, since there doesn’t appear to be a way to record multiple instances of the awards, just the number of merit badges earned, but a Scout can still earn and wear the multiple Palms.