Every morning while he ate breakfast, his dog would stare at him intently.
Literal
Every-morning breakfast eating during, his dog [topic-は] him [object-を] intently was-staring.
~ている間 ('while doing X') marks the time frame. じっと is a 擬態語 (state mimetic) for being fixed, motionless, and intent — the dog's unblinking gaze. 見つめる ('to gaze at, to stare') is more intense than 見る.