Background/Objectives: Targeted delivery of chemotherapeutic agents is a well-established approach to cancer therapy. Antibody–drug conjugates (ADCs) typically carry toxic payloads attached to a tumor-associated antigen-targeting IgG antibody via an enzyme-cleavable linker that releases the drug inside the cell. Aptamers are a promising alternative to antibodies in terms of antigen targeting; however, their polynucleotide nature and smaller size result in a completely different PK/PD profile compared to an IgG. This may prove advantageous: owing to their lower molecular weight, aptamer-drug conjugates may achieve better penetration of solid tumors compared to ADCs. Methods: On the way to therapeutic aptamer–drug conjugates, we aimed to develop a versatile and modular approach for the assembly of aptamer–enzymatically cleavable payload conjugates of various drug–aptamer ratios. We chose the epidermal growth factor receptor (EGFR), a transmembrane protein often overexpressed in brain tumors, as the target antigen. We used the 46 mer EGFR-targeting DNA sequence GR-20, monomethylauristatin E (MMAE) on the cathepsin-cleavable ValCit-p-aminobenzylcarbamate linker as the payload, and pentaerythritol-based tetraazide as the branching point for the straightforward synthesis of aptamer–drug conjugates by means of a stepwise Cu-catalyzed azide–alkyne cycloaddition (CuAAC) click reaction. Results: Branched aptamer conjugates of 1:3, 2:2, and 3:1 stoichiometry were synthesized and showed higher cytotoxic activity compared to a 1:1 conjugate, particularly on several glioma cell lines. Conclusions: This approach is convenient and potentially applicable to any aptamer sequence, as well as other payloads and cleavable linkers, thus paving the way for future development of aptamer–drug therapeutics by easily providing a range of branched conjugates for in vitro and in vivo testing.