Remote mining

Research the mineral, organic, or radiological technology until you get Robo-Miners, Robo-Farmers, and I forget what the radiological one is called. In stock, I believe its at level 10 that you get these; with Balance, its level 2.

Once you have the Robo-whateveryouwanttoremotemine, build a ship that has as many of those remote mining components as you can (obviously you'll need at least 1 engine room, and all the other stuff that goes onto a ship, such as master computer or bridge+crew quarters+living space) and then build the ship. Move the ship to the asteroid or planet/moon you wish to mine, and presto! You're remote mining it.

Remember only one ship can remote mine a sector at a time.