Why doesn’t the engineer count increase

These are what grant you XP and how much:

1. Alien Kill 30/60*
2. Regular Missions 60/90*
3. Storyline Missions 120/180*
4. Zero KIAs 20
5. Resist Psi Attack 10**
6. Successful Mindfray 20**
7. Successful Mind Inspiration 30**
8. Assist Mind Inspiration 30**
9. Successful Psi Panic 30**

*Additional XPs earned if the soldier has a rank of Lieutenant or less and kills any of the higher alien ranks (Ethereals, Sectopods, Elite Mutons, Sectoid Commanders, etc.) during the mission.

**Psionic actions.

Your soldiers have different requirements for promotions:

• Rookie 0
• Squaddie 90
• Corporal 300
• Sergeant 510
• Lieutenant 745
• Captain 1100
• Major 1560
• Colonel 2150
• Psionic* 0
• Specialist 50
• Operative 120

Source: http://ufopaedia.org/index.php?title=Soldiers_(EU2012)#Earning_XPs

Just to correct some mis-information in this question and responses: A classic Game (without the tutorial) starts you with 5 engineers + 5 scientists.

For both the workshop and laboratory, they require 6 engineers /scientists to unlock them for construction.

If you don't feel you can take the first 4 engineer (or scientist if your rushing the laboratory) mission that pops up, then rely on your second satellite to pick up an African or Asian country to get the +1 engineer bonus if you have set up in North/South America/Europe, giving you the required 6 engineers for the start of month 2. (For the lab, the opposite is true, place a satellite in North/South America/Europe).

I would say that getting the first 4 engineer mission is more helpful than controlling the panic levels however, allowing you to focus on something more beneficial such as the officer training school ^^