Is there such a thing as a dental implant that comes premade with an adjustable abutment head?

Many types: There are many different types of implants and prosthetic components on the market. Your surgeon and restorative dentist will use a system that they are comfortable with and that will satisfy the requirements of your reconstruction. Many times custom work is indicated due to implant angulation and existing space. Allow your dentist the latitude to make the best restoration s/he can.

Sort of: Some implants for very small spaces are one piece with their abutment. These abutment sections can be ground down a little if needed. Most other implants have stock abutmenhts that screw into place. These are of different sizes, angulations and shapes. There are also custom abutments that can be machined or cast if the anatomy is such that nothing off the shelf will work.

Yes: Dental implants are restored with a component part called the implant abutment. This is the piece that connects to the actual dental implant (usually via a small screw). These abutments can be custom made abutments or they can be 'stock' abutments. The biggest difference is cost and biomechanics of the restoration. I recommend custom abutments so that your tooth is made to fit your mouth best!

Yes, but why?: The best abutment is the custom abutment, where it fit the best with good emergence profile and proper adaptation for the implant crown support. It should be zirconia or nitric oxdide coating (look like gold plated) for better esthetic. Premade abutment are usually stock, generic, and ugly (in my standard).