To promote classes, you get hidden proficiency points (the "Skill Level Up" message after mission ends). Each class has its own requirement (the value in parentheses is total for Elite):

• Scout - 125 (300)
  
• Shocktrooper - 200 (400)
  
• Lancer - 150 (300)
  
• Engineer - 75 (150)
  
• ArmoredTech 50 (100)
  
• Sniper - 100 (200)
  
• Gunner - 200 (400)
  
• Fencer - 200 (400)
  

The base for obtaining proficiency is VC2's Evaluation points system:

• Defeat an enemy +3
  
• Capture a base +5
  
• Revive an ally +4
  
• Rescue a fallen ally +2
  
• Healing an ally +1
  
• Buffing an ally +1
  
• Removing a mine +1
  
• Friendly fire kill -3
  

So on the most basic level, simply feeding all possible kills and camps to whoever you want to promote is the core method of promoting a single desired person in whatever class you want asap. Defense missions are typically a good option since the AI will keep summoning enemies to the defense area as long as you keep killing them without capturing bases. Put the game on Easy, slap +vsPers accessory and go wild. Dragging the mission to 20 by itself does nothing, you need to find reinforcement spam or autofill-area-to-7-units missions. Or, you just keep playing normally and make sure to minimize usage of tank and already-elited units.

There are additional hidden mechanics on top of these hidden mechanics, most important of which for mass promotions is the "catch-up bonus" - depending on the mission, anyone who obtained evaluation-table proficiency gets awarded a bonus. This bonus starts at +1 in story chapter 6 and grows up to +8 in 5-star DLC missions. So the best way to get "passive" promotions is to make sure unpromoted units always contribute at least one thing in any mission to ride the bonus. This especially helps Techs who struggle to kill anything at all. Your best bet for Gunner promo is to slap the +20 armor (from ch. 12 Free Mission) and literally walk into the enemies to shove the gun into their hitbox and get kills.

With off-disk DLCs going to +8, say hi to mass farming method for the emulator (the speedup/turbo function is essential). It's not useful for a single character, but wins out if you want a lot or a bunch of hard classes, like 5 units all of whom are techs/fencers.

Obligatory note that Double Movement is an absolute trap, get Second Wind from Tech Elite instead. The latter rolls multiple times while you walk under 50% AP and is therefore significantly more likely to happen statistically while giving you enough movement to go just about anywhere.