Learn French: Notre chef parle à un client. - Our manager is speaking to a client.

Even though "our" means there are multiple owners, there is only one manager (chef). In French, the possessive word matches the thing being owned, not the owners. If we had multiple managers, we would use nos chefs.

If the manager was a woman, would notre change?

No, it stays exactly the same! Unlike mon and ma (my), the word notre (our) is used for both masculine and feminine singular nouns.

Doesn't chef just mean a cook?

It can, but in French, chef is a general word for a boss, manager, or leader. A restaurant cook is specifically a chef de cuisine, but in most workplaces, chef just means the boss!

Why does the French sentence use parle to mean "is speaking"?

French doesn't have a specific tense for "is doing" something right now. The simple present tense verb parle can mean either "speaks" or "is speaking" depending on what makes sense in English.