Only your professor can tell you that. Have you talked to your prof?
In my own case, my undergraduates often use the university dossier service or Interfolio, so I only write one or two generic letters (one for academia, one for industry, for example). I do suggest that if there is a job or position that is absolutely critical for them, that I'll write a personalized one. Or if it's a job/department where I know someone, I'll also offer to write personally.
However, that being said, what usually happens with American grad admissions is that you (the candidate) enter the names and e-mail addresses of your referees into the application system. It automatically generates a request to us (by e-mail) to fill in a quick summary (ranking) of the student and then upload a letter. In these cases, I usually do a quick customization.
In any case, the best judge of what to do is the professor. Show them a list of the schools you want to apply to and ask for their advice. You can suggest that you're willing to use a dossier service to take the burden off of them.
For my grad students, I almost always write personalized letters as there is a higher expectation there. Given the tanking of the job market, and some students applying to 60+ schools, this can become burdensome.
I assume that you've done all the other things to make things easier for them: i.e., create a spreadsheet with each department's name, discipline, mailing address, e-mail address, deadline date, and misc. notes, so that the prof can semi-automate the customization process. It's important for you to note which department you're applying to as the automated letter-request doesn't always note that, and it makes it difficult for me to know how to customize the letter, so I use a generic one.
The fun undergraduates are the ones who apply to security clearance level positions. Then I get to have a nice interview with the FBI or Secret Service about the student and their ethical/moral behavior. The feds always super polite and interesting people to talk to.