About 25 years ago I made a similar reflector and a small tripod, using a piece of thin stainless steel that I salvaged off of a baby posing table the studio was replacing with a new one. It was about 36 inches wide by 28 inches from front to back, and it already had MULTIPLE, tapped 1/4 x 20 N.C. threaded fixtures welded to the underside of it! I used it for many years. I guess it was 1/8 inch thick steel, and I painted one side white, and left the other side its natural stainless steel color. I used it as what J. Barry O'Rourke's book How To Photograph Women Beautifully called the "under-chin reflector", and it worked superbly on a regular, inexpensive tilt-style tripod head! Worked perfectly! The mounting hole was right in the center of the sheet of steel, and when cinched down, the rubber-covered mounting on the tripod's tilt head area fit tightly, and it supported and kept the sheet right in position. A tilt-head tripod head was actually the perfect way to mount the doggone thing!
How to Photograph Women Beautifully: Professional Techniques for Creating Glamourous Pictures: J. Barry O'Rourke: 9780817440046: Amazon.com: Books
A solid metal or wooden reflector, or even a piece of foam-core board (aka "art board"), or even a piece of formica can work wonderfully this way, especially for clamshell type lighting setups, where the main light is overhead and shining down, and the reflector is placed right around the chest level on headshots. The main light creates a nice catchlight at the top of the eyeball, and the reflector fills in shadows under the chin, and also creates a very nice, defining catchlight across the bottom of the eyeball, which adds a huge dimensional "cue". An overhead , high-positioned light and a just-out-of-camera-view under-chin reflector is STILL a staple of the headshot/beauty shooters...you'll see it on America's Top Model and so on. It just "works".