Can we measure the speed of light in one direction?

Yes, it is often assumed that Rømer measured the speed of light in one direction. It may seems trange, but Rømer velocity is also the velocity obtained under the tacit assumption of the equality of the speeds of light in opposite directions. The fact of the matter is that Rømer and Cassini were speculating about the movement of Jupiter’s satellites, automatically assuming that the observers’ space was isotropic.

The Estonian - Australian physicist Leo Karlov showed that Rømer actually measured the speed of light by implicitly making the assumption of the equality of the speeds of light back and forth.

L. Karlov, “Does Roemer's method yield a unidirectional speed of light?” Australian Journal of Physics 23, 243-258 (1970)

Also:

L. Karlov “Fact and Illusion in the speed of light determination of the Roemer type” American Journal of Physics, 49, 64-66 (1981)

Some reflections on the one-way speed of light are here.

Another interesting method to measure one - way speed of light that you may discover soon or later was so - called Double Fizeau Toothed wheel. That is two toothed wheel attached to opposite sides of long rotating shaft and a beam of light between the teeth. This method was employed (probably without proper due - diligence) by S. Marinov and M. D. Farid Ahmet.

However, Herbert Ives in his 1939 article "Theory of Double Fizeau toothed wheel" predicted that outcome of the measurement will be exactly c due to relativistic twist of the rotating shaft.

With two modern atomic clocks on well separated mountains, one observer could record the time when a short pulse passed his clock, and the second observer could record the time when it arrived at his. You might want to synchronize the clocks at a mid point, and position them at the same altitude.