The Egg Hammer is the first boss you'll face, a relatively simple introduction to boss mechanics. Dr. Eggman pilots a small mech equipped with a large hammer.
Phase 1: Hammer Swings
Eggman will swing his hammer in an arc, attempting to hit you. He'll then pause briefly. This pause is your window of opportunity. Jump over the hammer swing and attack Eggman's cockpit from above. Sonic's Homing Attack is perfect here. Tails can fly over the hammer and easily hit Eggman. Knuckles can glide over the hammer and punch. Amy needs to time her Hammer Jump to get over the swing and then hit Eggman. After a few hits, Eggman will retreat slightly.
Phase 2: Ground Pound
Eggman will occasionally perform a ground pound, creating a small shockwave. Jump to avoid the shockwave. The pattern remains similar: dodge the attack, then strike the cockpit. The arena is flat, so maintaining rings is easy. Keep attacking the cockpit until the Egg Hammer is destroyed. It takes about 8 hits to defeat.