Learn Farsi: پر کردن خندق به عهده‌ی ماست، ولی جیره‌بندی توسط پادشاه انجام میشه. - Filling the moat is our responsibility, but the rationing is done by the king.

The phrase به عهده‌ی literally means "on the shoulder/responsibility of". Here, it's followed by the pronoun ما (we/us) and the shortened "is" copula ست (st), creating به عهده‌ی ماست ("is our responsibility"). You could absolutely use بر عهده‌ی ماست instead; به and بر are interchangeable in this structure, though بر can sound just a little more formal.

Why is the infinitive پر کردن used at the beginning of the sentence? Does Farsi use infinitives like English gerunds ("-ing" words)?

Yes, exactly! In Farsi, the infinitive form (ending in -ن) is commonly used as a verbal noun or gerund. Here, پر کردن translates to "filling" and acts as the subject of the sentence. Notice that it connects to its object, خندق (moat), using an ezafe: پر کردنِ خندق (filling of the moat).

How does the passive voice work in the second half of the sentence with انجام میشه?

The active compound verb is انجام دادن (to do / to accomplish), but here we see its passive counterpart, انجام شدن (to be done), conjugated in the present tense as انجام میشه. When you need to specify who performs the action in a passive sentence, Farsi uses the word توسط (by / by means of), resulting in توسط پادشاه انجام میشه (is done by the king).