Showing torque on a magnetic moment is μ x Η

I'm a retiree who has decided to embark on a physics self-teaching project. Basically I want to relearn the physics I once knew but never needed, as well as learn a bunch of stuff I never took in school (QFT for instance).

I'm starting with Messiah's "Quantum Mechanics", which turns out to be a really good choice as his exercises go through a lot of classic calculations, making them really interesting as well as requiring me to also refresh my classical mechanics and electricity & magnetism at the same time.

So this question comes up in an exercise asking me to derive the equation of motion for a magnetic moment μ in a magnetic field H, then to solve it and show it precesses with the Larmor frequency.

The second half of that is easy. I know once I get an expression for the torque T in terms of the angular momentum L, I just solve T = dL/dt for L(t). The issue is in the first half.

Looking lots of places, I find the expression T = μ x H given without explanation, including in my old E & M text. But Messiah wants you to start with the energy U = -μ * H = -μH cos(θ).

My thought was that energy gives you a force F = -∇U. Then take something x F for the torque. But I'm stuck on the details of that. Taking H as the z axis, U depends only on θ so the (1/r) dU/dθ term is the only term in ∇U. But what is r here? And the same question arises with setting T = r x F.

I'm obviously forgetting something really elementary here and I may hate myself when I see the solution.

Maybe I should just assume the magnetic moment arises from a current loop of some fixed radius r? Then the r's cancel out and I don't care what it is?