。
She harbors the lofty ambition of becoming a great actress.
Literal
She [topic-は] great-actress [to-に] let-me-become [quotative-という] aspiration [object-を] is-cherishing.
Three patterns to track. 大女優 ('great actress') is a productive 大~ noun-prefix forming superlatives — 大成功 ('great success'), 大事件 ('major incident'). The construction X+になろうという+Y is built on the volitional 'X-になろう' ('let me become X') treated as quoted intent via という ('the [intent of] X'); と + いう ('say / speak of') here packages a goal-statement so it can modify a noun. 大望を抱く ('cherish a great ambition') uses 抱く ('to embrace, hold, harbor') in its abstract sense — applied to feelings, hopes, or doubts. The reading here is いだく (literary), not だく (physical).