Learn Farsi: شهردار در حال تماشا کردن جرثقیل است. - The mayor is currently watching the crane.

The phrase dar hâl-e (meaning "in the state of") must be followed by a noun or an infinitive acting as a noun. With compound verbs, you simply use the full long infinitive form. Here, that is tamâshâ kardan (to watch), giving us dar hâl-e tamâshâ kardan ("in the state of watching").

Why doesn't the object "the crane" (jarsaghil) have the specific direct object marker râ after it?

Because dar hâl-e forces the action into a noun phrase, the grammatical rules change. Instead of acting as a standard direct object that might take râ, jarsaghil connects to the infinitive using an Ezafe. You pronounce it tamâshâ kardan-e jarsaghil, which literally translates to "the watching of the crane".

The lesson mentions we can use a verbal noun instead of the full infinitive. How would that look in this sentence?

For many compound verbs, you can simply drop the kardan (to do) and use the noun half of the compound. Here, you could say dar hâl-e tamâshâ-ye jarsaghil ast. This is very common and often sounds much more elegant in formal Persian than using the bulky full infinitive.

What are the building blocks of the word shahrdâr (mayor)?

It is built from shahr (city) and -dâr, which is the present stem of the verb dâshtan (to have, to hold). So, a is literally a "city-holder" or "keeper of the city." You will see this suffix used all the time in Farsi to create titles and occupations.