To maintain organisational growth in a turbulent environment, organisations must build highly effective learning systems to innovate and develop from threats and stressors. This has been termed by Taleb (2012) as 'antifragility'. This paper explores the benefits of applying a systems approach to service delivery design in order to build an 'antifragile' organisation that can learn from disruptions. Two exploratory case studies were conducted in the UK insurance sector using in-depth interviews supported by documented evidence. The findings of the case studies analysis suggest that systems approach expressed as the Vanguard method (Seddon, 2003) is likely to enhance organisational 'antifragility' by promoting a multilevel driver for learning from stressors. These levels being: 1) the macro level of clarity on the system due to the continuous analysis of customer demands received; 2) the meso level of organic structure of work place where effective learning-centred teams are built; 3) the micro level of employees' engagement with work and readiness to learn. This paper represents an early effort to explore the dynamics of how organisations can go beyond resilience by discovering how to develop the capacity to learn from stressors in order to flourish.