Related answered questions: Chalcedonianism is a moderate between Nestorianism and Monophysitism?, What does it mean that the two natures of Christ cannot be separated?, and Does the Chalcedonian definition mean Christ has two minds?
Related unanswered question: How would miaphysites approach monothelitism versus dyothelitism?
I know the Sixth Ecumenical Council affirms the orthodox position of the two wills of Jesus and rejects monothelitism. And the Chalcedonian definition states
One and the Same Christ, Son, Lord, Only-begotten; acknowledged in Two Natures unconfusedly, unchangeably, indivisibly, inseparably; the difference of the Natures being in no way removed because of the Union, but rather the properties of each Nature being preserved, and (both) concurring into One Person and One Hypostasis; not as though He were parted or divided into Two Persons, but One and the Self-same Son and Only-begotten God, Word, Lord, Jesus Christ.
My understanding is that the indivisbility of the two natures means we can't attribute particular adjectives or actions that apply to the Person of Christ to the individual nature, though that may be its source. Such as, it would be incorrect to say that "Jesus' human nature died on the cross, but his divine nature did not." Or even "Jesus' divine nature is omnipotent, but his human nature is not." Though we may know that the source or origin of his omnipotence from his divinity, we cannot attribute omnipotence to Jesus' divine nature but the person as a whole. It seems the Chalcedonian definition supports this, unless "the properties of each Nature being preserved" implies the opposite of that.
By "attributing" a property, or perhaps "identifying" a property to be of one nature, I am referring to the accuracy of statements such as "Jesus' human nature slept, but Jesus' divine nature was always awake" (because "God never sleeps", Psalm 121:4). If we can't attribute a property to a particular nature, then we must stop at saying "Jesus slept" (the person of Christ) and cannot say anything in particular about what each individual nature experienced, though we perhaps could say that the origin of Jesus' ability to sleep is from his human nature. The same applies with Jesus' omnipotence. Perhaps we could say that the origin of Jesus' omnipotence is his divine nature, but we can only say that Jesus is omnipotent (not saying 1/2 of his natures is omnipotent).
The communication of properties between Jesus and God seem to come into play here, as well. If we can say that Jesus slept, then that means God slept. But, if only His human nature slept while His divine nature was awake, then perhaps we could escape concluding that God slept? Except that the Bible seems to be denote the person of Jesus with actions or adjectives, rather than an individual nature.
It seems like this being the case, we could only attribute the will of Jesus to the person of Jesus, and not either individual nature when the two are inseparable. It seems like the same arguments apply for His wills as for various adjectives. "Jesus wouldn't be 'fully' human if he didn't have a human will." "Well, Jesus wouldn't be 'fully' human if he wasn't limited in knowledge, and yet he possesses omnipotence." Or something like that. Take the claim "Jesus can't be fully human without a human will;" why can't Jesus be fully human because He has a will as a person? As in, a will that is attributed to the person of Christ rather than to his individual human nature. I don't get how that wouldn't fulfill the "fully human" requirement. It seems that to say otherwise is just based on how we define what "human" is (which of course would be important).
If we can't attribute adjectives or actions to either individual nature, why can we attribute wills to the individual nature? How is that not separating the two natures that should be indivisible? From a typical orthodox Trinitarian view, I want to know how this doctrine is properly formulated in light of these concerns, whether through Church creeds or early church fathers or theologians of the day or through someone's explanation.
Edit: suggested from comments below, how do do we know it is acceptable to attribute a property to one nature and not the other given that the two natures are inseparable? What does it mean for them to be inseparable if you can identify properties of each individual nature rather than the Person?