。
Let's show off our cooking skills.
Literal
Arms [object-を] wield cooking [object-を] do-[volitional-しましょう].
腕を振るう ('to wield one's arms, exercise one's skills') is an idiom meaning 'to show off one's skill, display one's talents' — the 'arm' here metaphorically represents one's ability, and 振るう ('to wield, exercise') is the formal/literary verb for exercising power or skill. Commonly paired with 料理を振るう ('to show off cooking'). Volitional しましょう invites joint action.