use a hover vehicle. In the past when building a base in the side of a mountain ive built an ugly small one with a couple mechanical drills. You can lock the hover height and move more slowly to get a relatively flat ground level. The hand drills are quite a bit less effective than the vehicle drills on purpose in RE to encourage you to make a vehicle. It takes a bit of time to dig out a space but it works pretty well. You may want to look at the guides section for emperyion to find a guide that lists all the hot keys as not all of them are listed in the game itself. Share a picture of the base after!

yea i know the vehichel part is better, but sometimes you want to do it "on foot"