Answer: Combo Emboss!
There are 3 levels of embossing available to you. In some cases, your design will dictate which kind of die you should use. In others, your budget will lead the way. The larger the embossed area and the more complex the design, the more costly the die.
In the case of a combo emboss (aka “combination embossing” or “foil embossing”), the process of embossing and foil stamping is accomplished in 1 print run with the use of a combination die. The combination die has a cutting edge around the perimeter to cleanly break the excess foil away from the embossed area. (Check out a gorgeous example of this technique right here.)