I was talking to some people on the bridge about this. They said that it might be because of the addition to the voltage, thats what makes the clicking noise...
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When your phone is just sitting there, not doing anything, it's running at about 20v. It goes up to around 50v when someone rings you, and drops to 10v when you answer. When you beige box, you are doing the same thing as picking up a phone in their house.
You remember when your little sister would try to listen in on your calls, but she was too stupid to know that when she picked up the phone you could hear it click? Well, the same thing is happening (in fact, that's how pulse dialling works). Every time a phone picks or hangs up, it changes the line voltage.
The "slowly turning up" method, to me, would be even worse. The voltage would be changing constantly, resulting in far more impulse noise than one swift pick-up would.
I don't think there is any perfect way to do this without picking up before they do. But ask around, someone else may have something to help you =D.
