First you must create an enhanced weapon for your mob so you can direct him in battle. Talk to blonde (Klein) in science and he will make if you supply ingredients. Typically it is 10 ymir (the skulls) and 3-5 weapon drop from mob. Check your mob status screen for type. The types are tooth, tusk, claw "scissors"(pincers) or instrument. First 3 are common, others are rare. Once you have weapon, summon your mob and step on green hexagram in office.
I think doing above is prerequisite to evolution.
Evolution is done by getting mob to 20th level, having 20 ymir and entering evolve chamber. That chamber is accessed by going through the double sparkle on cabinet in office. There are 8 (edit: 6)corridors for the mobs that were the 1st 8 level bosses. First hex in row will evolve to 2nd stage then disappear. Get to 20 again and gather 50 ymir to evolve again on 2nd hex.
NOTE: DO NOT KNOW IF YOU CAN EVOLVE IF OVER 20TH LEVEL. Several of mine were well above 20 and wouldn't but I do not know if mob at 21-25 may work. I would suggest getting them to 20 and porting out of dungeon.
EDIT: THEY CAN EVOLVE EVEN IF WELL OVER LEVEL 20. the one I tried first was my level 1 boss and he and 2nd level boss must not be eligible. there are 6 corridors not 8.
ALSO: I
think my first 2 basic evolves cost 20 ymir, but so far the rest are costing me 30 ymir. Have not done any more secondary evolves yet.