Spectral estimation provides key insights into the frequency domain characteristics of a time series. Naive nonparametric estimates of the spectral density, such as the periodogram, are inconsistent, and the more advanced lag window or multitaper estimators are still too noisy. One popular solution is to semiparametrically model the log spectral density using basis functions. The basis coefficients, and hence the spectral density, are usually estimated using penalized least squares with the log periodogram. Using an L1 penalty allows for simultaneous estimation and model selection, but the choice of family of basis functions is often limited by the optimization algorithm used. We propose an L1 penalized quasi-likelihood Whittle framework based on multitaper spectral estimates and present an alternating direction method of multipliers algorithm to efficiently solve the optimization problem. We develop universal threshold and generalized information criterion strategies for efficient tuning parameter selection that outperform cross-validation methods. We demonstrate how well our methodology performs on simulated series and to the spectral analysis of electroencephalogram data.