Make a unit, use self in drivers.
Recommend first making a driven unit.
For this example I have used a plane added a solidify modifier and driven the modifier thickness. Similarly could use a cube and drive object scale, after moving the origin to the bottom face of the cube
If we check use self in a driver it is the object being driven. In this case it is the solidify modifier, to get the plane instead, we use self.id_data (The blender ID object that owns the modifier.)
The matrix world of the empty is added as the variable mt Every object has a matrix world that contains the global transform of the object. (Similar to using the world location transform channel)

The driver expression
-max(3 - (self.id_data.matrix_world.translation.xy - mt.translation.xy).length , 0) * mt.translation.z / 3
This part of the expression
max(3 - (self.id_data.matrix_world.translation.xy - mt.translation.xy).length , 0)
confines the result to be within 3 units of the 2d xy circle around the plane origin. If the empty goes outside this it will have 0 influence.
It is then mulitiplied by the global z location of the empty, and divided by 3 to normalize it such that at best the solidified plane can touch the empty.
I've assumed for simplicity the plane will sit on the z=0 plane, (other wise would need to subtract the planes global z coordinate)
Finally it is negated since we are solidifying against the plane normal. (Flip in edit mode to use +ve)

Ok, if happy with out unit, add an array modifier to it, apply the modifier, in edit mode, P separate by loose parts. In object mode Object > Set Origin > Origin to Geometry to make each plane have its origin at its center, not that of the original unit.
Quick example with 5 x 5 applied array mods
Extending.

Have arbitrarily chosen 3 as the XY radius of influence. Could add a custom property "radius" to the empty add another single property variable in the expression in place of 3. Can now also animate the radius of influence of the empty.
Another tip is if the expression gets too long to see in the driver field, try using the python console to check them out.
>>> mt = D.objects['Empty'].matrix_world
>>> self = C.object.modifiers['Solidify']
>>> (mt.translation.xy - self.id_data.matrix_world.translation.xy).length
1.9968496634438566
>>>