His boss is Ernestine. You send the note to Ernestine so she can put it in her file on him and maybe give him an 'attaboy. A letter of recommendation is slightly different. If Frank is changing companies, a similar letter could recommend him to the prospective employer. Good service should be recognized, it's the right thing to do .
Official recognition can give someone an emotional boost , along with their friends and associates.
If you haven't been around more than a few months at the company, it would be good to wait until you're settled in and have a feel for Frank's work there over time. If you don't know what Frank's official job description is, ask Ernestine. OK, so you know what Frank's job is. If he does his job really well, that's something. Frank is a helpful, compassionate guy.