The European Spallation Source (ESS) is a pulsed spallation neutron source currently being built in Lund, Sweden. At full power and beam energy, ESS will accelerate protons to 2 GeV at 5 MW (average pulse current of 62.5 mA, pulse length of 2.86 ms, and repetition rate of 14 Hz) onto a rotating, helium-cooled, tungsten target. This target is in a heavily shielded structure referred to as the target monolith, which is located inside the ESS target building. Directly on top of the target monolith shielding is a room referred to as the connection cell, which contains many important utility connections and control systems. Many connections from components in the target monolith, e.g., cooling water and electrical connections, exit the monolith into the connection cell and pass through the walls of the connection cell to reach other areas of ESS. When the accelerator is operating at ESS, the connection cell will be off limits to humans. This paper evaluates the prompt dose rates to several important materials in the connection cell and suggests an algorithm for evaluating the suitability of materials that might be used in the connection cell.