Do you mean English when you say "British"? In 1775 probably one third of the Continental Army were not born in America.

John Paul Jones was Scottish and arrived in America in 1776.

Thomas Paine was a very recent immigrant to America, having immigrated from England in 1774.

Casimir Pulaski was Polish and fought against Russia when that country invaded Poland. He then joined the American Revolution as a cavalryman and distinguished himself in many ways.

Wilhelm von Steuben was a Prussian who was one of the most influential officers in the Continental Army. He was responsible for completely overhauling the way the Continental Army trained and drilled it's men. He's often credited with turning the Continental Army into a professional fighting force, though I think that's giving him too much credit. He definitely did standardize and revolutionize the training of the Army, and he was absolutely a critical figure.

Johann de Kalb was a German who ended up as a major general and was killed during the Battle of Camden.

Michael Kovats was a Hungarian who had served with the Hungarian cavalry and then with the cavalry of Frederick the Great. In 1777 he offered his services to Franklin and Franklin referred him to Pulaski. He was killed in 1779.

Andrzej Tadeusz Bonawentura Kościuszko was another Polish volunteer. He was an engineer who provided critical assistance to Nathaniel Greene. Kosciuszko also helped design the original fortifications at West Point.

Charles Lee was born in England. He served during the Seven Years War. After the war ended in 1763 he served in various armies in Europe. In 1773 he moved to Virginia. He was commissioned a general in the Continental Army and would prove to a major thorn in Washington's side.

Those are some of the more famous men who fought for the American side during the Revolutionary War, but like I mentioned earlier a large portion of the American Army wasn't native born American. To quote from 1775: A Good Year For Revolution:

"The ethnic makeup of British North America in 1775 also complicated Patriot enlistment and mobilization. Over one third of the colonists taking up arms for "the rights of Englishmen" were German, Dutch, Irish, Scottish, or Scotch-Irish."

There were enough Germans living in Maryland and Pennsylvania to provide a battalion of men who were German speakers. This battalion fought in almost every battle of the northern campaign.

So yes, there were lots of Europeans who came to America to join the cause. As for needing to recruit, Franklin really didn't need to do that. There were plenty of volunteers--in fact there were too many volunteers. When Lafayette came over, he came over with de Kalb and several other Europeans who were planning on acquiring commissions in the American army. Only Lafayette and de Kalb were offered positions, and Lafayette served without pay