What does "steep incentive contract" mean in the context of adverse selection? This term was mentioned in a slide and when I looked online, the only thing which I could find talking about this is this paper. 

For example,in the cost-plus-incentive contract

$$x(T, c) = a(T) - a'(T) × (T-c)$$

where $x$ is the contractor's fee, $T$ is the cost target, $c$ is a verifiable cost; $a(T)$ is decreasing and convex.

I was told that with smaller $T$ and hence higher $a(T)$ and $-a'(T)$, the incentive contract is steeper. Is the steepness represented by $-a'(T)$? What exactly is the interpretation of steepness of incentive contract? Does it mean more risk for the contractor? If this is the case, am I correct that the decreasing $a(T)$ motivates the contractor to take a smaller T? How is it translated to incentive constraint? That high-cost contractor would not recklessly take a small T?
"Steep incentives schemes" are synonymous with "high-powered incentives". A steep incentive scheme is thus one in which there is a large or high-powered performance incentive. This can take either the form of a small fixed payment and a (potentially) large bonus, or it can be a large fixed payment and a (potentially) large penalty for deviation.

Normally we think about using steep incentive schemes when we wish to have a separating equilibrium in which high-skilled (or risk-seeking) workers choose steep incentive schemes and low-skilled (or risk-averse) workers choose flatter incentive schemes. Here is one paper I'm aware of that discusses this. You will find more examples if you look at optimal contracts under risk-aversion. https://www.econstor.eu/bitstream/10419/30169/1/620692227.pdf

In your example, I believe the decreasing nature of $a(T)$ is capturing the movement from a low-cost (less risky) project to a high-cost (more risky) project. As the cost (risk) for the project increases, the principal will want a steep incentive scheme to ensure that only a high-skilled contractor will choose to perform the task. This is because we are shifting the weight away from $a(T)$ (the fixed payment) and towards $a'(T) \times (T-C)$ (the bonus/penalty) as $T$ increases. Moving the other direction, as the cost (risk) for the project decreases, $a(T)$ increases and the incentives have less power.