Analog-to-digital converters (ADCs) play a critical role in digital signal acquisition across various applications, but their performance is inherently constrained by sampling rates and bit budgets. This bit budget imposes a trade-off between dynamic range (DR) and digital resolution, with ADC energy consumption scaling linearly with sampling rate and exponentially with bit depth. To bypass this, numerous approaches, including oversampling with low-resolution ADCs, have been explored. A prominent example is 1-Bit ADCs with Sigma-Delta Quantization (SDQ), a widely used consumer-grade solution. However, SDQs suffer from overloading or saturation issues, limiting their ability to handle inputs with arbitrary DR. The Unlimited Sensing Framework (USF) addresses this challenge by injecting modulo non-linearity in hardware, resulting in a new digital sensing technology. In this paper, we introduce a novel 1-Bit sampling architecture that extends both conventional 1-Bit SDQ and USF. Our contributions are twofold: (1) We generalize the concept of noise shaping beyond the Fourier domain, allowing the inclusion of non-bandlimited signals in the Fourier domain but bandlimited in alternative transform domains. (2) Building on this generalization, we develop a new transform-domain recovery method for 1-Bit USF. When applied to the Fourier domain, our method demonstrates superior performance compared to existing time-domain techniques, offering reduced oversampling requirements and improved robustness. Extensive numerical experiments validate our findings, laying the groundwork for a broader generalization of 1-Bit sampling systems.