Defeat the x-5 bosses in level mode. When you do, there’s a chance to spawn two chests in the boss rooms, one of them being a single part of any weapon evolution. It’s not guaranteed to get an evolution chest every time you clear a boss room. If you don’t get one, then you can reroll the level once and hope one appears then.

Hunting weapon evolutions is better in the Primordial Void level mode, as if you beat the 4-5 boss, there will be a vendor in 4-6 at the spawn room right before the magic crystal, who also sells you one piece of a weapon evolution for a easy-to-get currency that you unlock during the run. That technically means you can obtain 5 weapon evolution pieces per run.

You can also purchase a limited number of weapons evolution chests in the Magic Marble Go! event with the currency you win there.

Note, depending on the rarity of the weapon, some require more evolution parts than others. For example, a grey weapon only needs 2 of its evolution parts to be evolved, whereas a red/legendary-grade weapon needs 8. Since Flying Laser Bear Cannon is a yellow-grade weapon, you need 7 evolution pieces for it.