BOI Tower Controller reported an aircraft executing the published ILS missed approach in foggy conditions passed too close to the Control Tower. Reporter stated aircraft passing close to Tower on this procedure has occurred on several occasions; and recommended a change to the procedure.

Narrative

I was working local control. Fog just moved in and the airport went IFR. Several aircraft were going around due to low visibility. Aircraft X on the ILS [Y RWY 10R] went around and I issued the published missed approach. I noticed the aircraft was slightly off course and looked like they made a right turn to the southeast a little early. I made a half joking comment to Ground Controller that they were going to hit us. He looked up at the Tower radar display and agreed. The fog was moving west to east and there was a small clearing front of us. We saw Aircraft X pass very close to the Tower. Closer than an Aircraft X should ever get to the Tower. The Ground Controller and I were cursing in disbelief. The supervisor was looking at some paperwork so he did not see Aircraft X pass close by. This has happened to me several times before on ILS go arounds but usually with smaller aircraft and I can't actually see them due to the fog. I have on several occasions heard aircraft passing very close to the Tower in ILS go arounds. I saw this one and it was [too] close to not report. I believe an aircraft will eventually hit the Tower one day.Change the ILS missed approach procedure to go straight out instead of intercepting the 113 radial outbound. I believe aircraft are going around and turning to intercept the radial early which points them right at the Tower.