If a valve stem is broken, can it be welded back together in field?
If the welded area goes through the packing, he packing will be destroyed on the first stroke.  If you weld it then chuck it on a lathe and true it up before you reinstall it, you probably have bar/rod on hand that you could use just to machine a replacement. Roller-Burnishing the machined surface both smooths it and puts the surface in compression so it is less likely to crack.  
If the valve is a rotary control valve there is a greater-than-small possibility that the stem is heat-treated (17-4 PH) which will be damaged in the heat-affected zone. You had better get a new, proper metallurgy stem on an airplane to you right now.

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