Fig.: to use a disproportionate force or expense to overcome a minor problem

The police response to the few peace demonstrators was absolute out of proportion; it was like shooting a mouse with an elephant gun

La reazione della polizia nei confronti del piccolo corteo pacifista è stata assolutamente spoporzionata; è stato come uccidere un moscerino con una cannonata

After little George got slightly hurt in a school soccer game, his mother withdrew him from any sport activities. It seems to me this is cracking a nut with a sledgehammer

Dopo che Giorgino si è leggermente infortunato in una partita di calcio scolastico, sua madre lo ha ritirato da qualsiasi attività sportiva. Mi sembra proprio voler uccidere una mosca con una zampa di elefante

Look, it's true that you have some issues at work, but resigning and moving to another country over this, would be killing a fly with a cannon

Senti, è vero che hai qualche problemuccio sul lavoro, ma licenziarsi e trasferirsi in un'altra nazione per questo, sarebbe come uccidere un moscerino con una cannonata

Sledge-hammers are large iron hammers, and this phrase wasn't first saw the light of day in 1850s America. A sledgehammer to crack a nut is one of the many versions of the phrase, the others having faded into disuse. Pretty well anything which is small and easy to squash has come verbally under the hammer, particularly nuts (peanuts, walnuts, nuts) or insects (gnats, flies, mosquitoes etc.). The first to fall victim was the humble fly, as in this piece from The Gettysburg Compiler, June 1878: "Don't worry over little ills of life. It is like taking a sledge hammer to kill a fly."
Nuts came into the picture a little later, specifically peanuts, and the first example of that in print dates from the 1960s.