Creating a map of energy from gravitation measurements There are some devices which used supercooled falling atoms to measure gravity in order to weakly find large regions of different density.
Would it be possible with current technology to measure gravity precisely enough to create a dense map (meter / sub meter resolution) of energy (mass) in the local area? How would this be done?
 A: There are methods that do map density (mass) variations in the earth. Not just on a local scale, but literally on a global scale. Perhaps the most famous method (using satellites) is talked about below$^1$.  The method you have stated can and will be used on space craft as well$^2$.
$^1$ Since the gravitational force is calculated using the mass a given material has, the more mass it has, the stronger will be its gravitational pull. Because the earth is not uniform in mass density, there will be perturbations in the Earth's gravitational field. Normally, scientists and engineers would construct these "gravity maps" by using local land measurements and more recently, remote sensing. However, those measurements weren't accurate enough to capture the slight changes over local areas (and larger areas).
In 2002, an international team of engineers and researchers developed the Gravity Recovery and Climate Experiment (GRACE) mission. From this link,
"The Gravity Recovery and Climate Experiment ( GRACE) was a joint mission of NASA and the German Aerospace Center (DLR). Twin satellites took detailed measurements of Earth's gravity field anomalies from its launch in March 2002 to the end of its science mission in October 2017."

The two GRACE satellites fly with a distance of approximately $220 km$ between each other. A microwave ranging system monitors distances between each satellite to within one micrometer. The scientists can therefore map gravitational force anywhere in the Earth by measuring tiny changes in distances between these two satellites (as each of them changes speed in response to varying gravitational force).

This image shows gravity (density) perturbations in the Earth, courtesy NASA.
Dark blue shows areas with lower than normal gravity, such as the Indian Ocean (far right of image) and the Congo river basin in Africa. Dark red areas indicate areas with higher than normal gravity. The long red bump protruding from the lower left side of the image indicates the Andes Mountains in South America, while the red bump on the upper right side of the image indicates the Himalayan mountains in Asia.
As can be seen, Earth's mass is distributed between various landforms and features, like mountain ranges, oceans, and deep sea trenches. All have different mass, which creates an unevenly distributed gravitational field.

$^2$ The goal in such measurements (and indeed all measurements) is always to obtain greater precision, so NASA teamed up with the company AOSense Inc which uses your example of supercooled atoms but with a different approach.
The idea behind using supercooled atoms, uses Cesium atoms sealed in a vacuum near absolute zero, and the atoms are insulated from outside influences that may affect the measurements, such that the only force acting on them is gravity.
But the new approach uses a smaller type of sensor (original ones were too large) that is small enough to fit on a satellite. But even though it is of smaller size, there is no sacrifice on sensitivity. Actually, during its testing, the device reported different gravitational forces after the researchers came back from lunch break. The device was actually detecting the added mass of food in their stomachs.
Once this device is up in space, it is believed that the sensor will map the Earth’s gravity $10 \times$ more accurately than with the GRACE system above, and also at $4 \times$ the spatial resolution.
