gouge out to a "V" and fill in. you could weld a 2" plate this way , remove all slag between each pass and it should be a good weld. I am not shure about additional passes without gouging out. it seems to me the arc will only penitrate to the same depth the second pass, but on the second pass you would be trying to penitrate through the first bead of weld. rite... If you can not get full penitration form each side that gouge out and do multipal passes. by full penitration from both sides I mean at least half way so they meet in the middel.
rember Heat is the enimey of metal so gouging and dooing three passes with lower amps will be better than one HEAVY pass for the metal.
As for MIG welding 1 1/4" ........ HMMMM ware do I start. if it is not going to take a lot of stress, maby something like a spacer in compression. but if you are going to load it up with stress I can tell you ware it will let go. I would gouge out 1 1/4" from both sides even if I was useing a stick welder. you can get it plenty strong but honestly it should be welded all the way through or it is a significant weak spot. rember to remove slag betweel passes if you use a flux core.
oh yes,, you should gouge out leaveing enough in the center not removed that will be approximetly the amount you can get full penitraton from both sides. so if you are going to get 1/4 inch penitration you shoudl leave about 1/2 inch. if it needs to stay flat tack a spot or two and cool. repeat till well tacked on both sides, then run short beads and cool repeat till full. yes it would take a lot of time on a thick plate.
hope I did not confuse thinges even more and this is just my opinion after 20+ years of welding.