It is true that when both principal and agent are risk neutral, the first best can be obtained despite asymmetric information. You should refer to a textbook, such as MWG (ch.14), for the technical details of such models. I'll give an intuitive explanation here. The intuition of the result lies in optimal risk sharing between the principal and the agent.

With a risk neutral principal and risk averse agent, optimal risk sharing requires that the principal bear all the risk and the agent bear none. Hence, first best is generally not attainable in such a setting because, with asymmetric info, the principal cannot condition pay on effort but on outcome instead. As a result, the agent would have to bear some risk in equilibrium. To satisfy the agent's participation (or individual rationality) constraint, the principal needs to compensate the agent for his risk taking, thereby reducing the total gains from the contract.

When both parties are risk neutral, there is no cost for the agent to bear risk. Therefore, optimal risk sharing between the principal and the agent is attained, which results in the same outcome as the first best.

Let's view the problem of contracting with moral hazard from a slightly different perspective. Suppose, instead of trying to monitor the agent's effort, the principal simply sells the shop to the agent for a fixed fee, and let the agent retain whatever profit it can make from the business. As the de facto owner, the agent would thus have incentive to exert the optimal level of effort. [This is the quote from Bolton and Dewatripont mentioned in @Hal_Incandenza's answer.] But how much of the fixed fee can the principal charge for transferring ownership?

Well, to respect the participation constraint, the fee can be at most the certainty equivalent of the agent. If the agent is risk averse, his certainty equivalent is less than the expected value of the business, hence the efficiency loss in the second best outcome. If the agent is risk neutral, his certainty equivalent is the same as the expected value of the business; thus there is no efficiency loss in the second best outcome.

It is true that when both principal and agent are risk neutral, the first best can be obtained despite asymmetric information. You should refer to a textbook, such as MWG (ch.14), for the technical details of such models. I'll give an intuitive explanation here. The intuition of the result lies in optimal risk sharing between the principal and the agent.

With a risk neutral principal and risk averse agent, optimal risk sharing requires that the principal bear all the risk and the agent bear none. Hence, first best is generally not attainable in such a setting because, with asymmetric info, the principal cannot condition pay on effort but on outcome instead. As a result, the agent would have to bear some risk in equilibrium. To satisfy the agent's participation (or individual rationality) constraint, the principal needs to compensate the agent for his risk taking, thereby reducing the total gains from the contract.

When both parties are risk neutral, there is no cost for the agent to bear risk. Therefore, optimal risk sharing between the principal and the agent is attained even if pay is based on outcome, not effort. This results in the same outcome as the first best.

Let's view the problem of contracting with moral hazard from a slightly different perspective. Suppose, instead of trying to monitor the agent's effort, the principal simply sells the shop to the agent for a fixed fee, and let the agent retain whatever profit it can make from the business. As the de facto owner, the agent would thus have incentive to exert the optimal level of effort. [This is the quote from Bolton and Dewatripont mentioned in @Hal_Incandenza's answer.] But how much of the fixed fee can the principal charge for transferring ownership?

Well, to respect the participation constraint, the fee can be at most the certainty equivalent of the agent. If the agent is risk averse, his certainty equivalent is less than the expected value of the business, hence the efficiency loss in the second best outcome. If the agent is risk neutral, his certainty equivalent is the same as the expected value of the business; thus there is no efficiency loss in the second best outcome.