Why don't you need et with lechol, is it because it's not a direct object?
Infinitive: a special verb form that is unchanged for gender or plural, and has an abstract meaning. In English: to go, to be, to squeeze.
The infinitive a Form of the infinitive The infinitive cannot be inflected. Whether one is addressing males or females, one person or many, it is unchanged. The infinitive’s distinguishing mark is a prefixed Lamed ל ,
To eat: (kind of like The different ways you'd use "to eat, but couldn't say to Eatens, to eaten-ing, aten etc). Instead you it would depend on the tense and the words around it, eat, eating, ate, eaten, ..
(To write: written, writing, wrote, writes. To drive: drive, (was, will be )driven, (will be, has been, had been, planned on), driving, drives, drove,). Etc.
https://www.pealim.com/dict/30-leechol/ is The conjugation chart for l'ochel. I'm learning too, most of the above is from COLLOQUIAL Hebrew or modern Hebrew Grammar.
Is "pil" feminine? Because it shows the feminine infinitive of "lekhal."