There's an enrichment available in Clay called "Find Employee Headcount by Criteria". You could use this, and within the Job Title Keywords, supply the relevant terms for finance roles. (You would need the company's Domain or LinkedIn Profile URL as a starting point.)
For recent hires, you can use the same integration and include all relevant keywords to capture specific job titles ("CFO", "Chief Financial Officer", "VP of Finance", "Vice President of Finance"), but that would only give you a number, not recency.
I'd recommend the "Find People" search, where you can specific a "Maximum months in current role".