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The study of characteristic gluing was initiated by Aretakis, Czimek, and Rodnianski (2021), and has since proven to be an effective tool for constructing solutions to the Einstein equations. A remarkable example of this is the work of Kehle and Unger, who used it to construct solutions to the Einstein–Maxwell-charged scalar field system undergoing gravitational collapse to form an exactly extremal Reissner–Nordström black hole in finite time, disproving the so-called third law of black hole thermodynamics, and challenging the intuition that had prevailed for decades in much of the theoretical physics community. All these works focused on the Einstein equations with vanishing cosmological constant. In this talk, we discuss how these techniques can be extended to include a positive cosmological constant and applied to construct solutions to the Einstein-scalar field system in which an exactly extremal Schwarzschild–de Sitter black hole is formed. Time permitting, we will also discuss the dynamical formation of the related extremal Schwarzschild–de Sitter white hole and Nariai spacetimes.