A laser of a known frequency is pointed into space up from the north pole. A frequency analyzyer on a satellite is far away north of the earth's vertical axis. Is it possible from the gravitational redshift of the laser to calculate the gravity (and therefore mass) of the earth?
Precise communications with satellites (such as those responsible for providing Global Positioning System data) include corrections for General Relativity, which means that corrections are needed to deal with the gravitational pull of the Earth (and the sun). These can be (and has been) turned around to give a measurement of the Earth's mass.
The first Doppler shift measurement of the Earth's gravity was actually a historically important test of General Relativity. The test was performed not using optical or radio transmissions but $\gamma$-rays, utilizing the extremely precise technique of Mössbauer spectroscopy. The experiment, done by Pound and Rebka in 1959, measured the gravitational redshift due to the Earth at about the 10% level of precision. (The precision was limited by a number of unforeseen systematic problems, such as needing to have the source and detector materials at almost precisely the same temperature, to avoid additional Doppler shifts due to the internal thermal motions of the materials.) You can still visit the vertical chamber where the experiment was conducted at Harvard University.