Learn French: Le chef envoie une facture pour ce compte. - The boss is sending an invoice for this account.

Envoie comes from the verb envoyer (to send). For verbs ending in -yer, the y changes to an i when it is followed by a silent e. This happens in the forms for I, you (singular), he/she/it, and they. So, il envoie uses an i, but nous envoyons keeps the y.

Doesn't chef mean a cook?

It can, but its main meaning is actually "boss", "manager", or "head" of any team or department. A restaurant cook is called a chef because they are the "boss" of the kitchen!

Why do we use ce for "this" here instead of cet or cette?

Compte (account) is a masculine word, so we use the basic masculine word for "this", which is ce. You would only use cet if the masculine word started with a vowel sound (like cet ami), and cette is only used for feminine words.

Could compte also mean a bill at a restaurant?

Not usually. Compte means an account, like a bank account or client account. If you want the bill at a restaurant, you ask for l'addition. An invoice for business goods or services, like the one the boss is sending in this sentence, is une facture.