Indium and silver are technologically important, critical metals, and in the majority of cases, they are extracted as a by-product of another carrier metal. The importance of indium has seen recent growth, and for technological reasons, these metals can be found in industrial residues from primary zinc production, such as the iron precipitate—jarosite. To secure the supply of such metals in Europe, and with the idea of a circular economy and the sustainable use of raw materials, the recycling of such industrial residues is coming into focus. Due to the low value of jarosite, the focus must lie simultaneously on the recovery of valuable metals and the production of high-quality products in order to pursue an economical process. The objective of this article is to give the fundamentals for the development of a successful process to extract the minor elements from roasted jarosite. As such, we use thermodynamic calculations to show the behavior of indium and silver, leading to a recommendation for the required conditions for a successful extraction process. In summary, the formation of chlorine compounds shows high potential to meet the challenge of simultaneously recovering these metals together with zinc at the lowest possible energy input.