Computer systems often provide hardware support for isolation mechanisms like privilege levels, virtual memory, or enclaved execution. Over the past years, several successful software-based side-channel attacks have been developed that break, or at least significantly weaken the isolation that these mechanisms offer. Extending a processor with new architectural or micro-architectural features, brings a risk of introducing new such side-channel attacks. This paper studies the problem of extending a processor with new features without weakening the security of the isolation mechanisms that the processor offers. We propose to use full abstraction as a formal criterion for the security of a processor extension, and we instantiate that criterion to the concrete case of extending a microprocessor that supports enclaved execution with secure interruptibility of these enclaves. This is a very relevant instantiation as several recent papers have shown that interruptibility of enclaves leads to a variety of software-based side-channel attacks. We propose a design for interruptible enclaves, and prove that it satisfies our security criterion. We also implement the design on an open-source enclave-enabled microprocessor, and evaluate the cost of our design in terms of performance and hardware size.