Without some major machinery, it would be a tough job. It also requires some practiced skill to do it effectively. The milk pitchers are made by a process called metal spinning by which a flat blank of material is placed in a machine resembling a wood lathe. The operator uses a handheld tool similar to a wood lathe tool except it isn't sharp, rather it is rounded. A force it then applied with the tool against the material to cause it to bend up to the required shape while spinning. The spout may be formed before or after the primary shaping.
Your first problem would be how to hold the pitcher. Some kind of jig/holder would be required since the force to shape the pitcher is not a light force.
Minor adjustments of small fractions of an inch/mm could be done using hand-held tools similar to metalsmiths tools. But it would be tedious and require some skill to do w/o ruining the pitcher totally.