We have measured electron-ion recombination for Fe⁹⁺ forming Fe⁸⁺ and for Fe¹⁰⁺ forming Fe⁹⁺ using a merged beams arrangement at the TSR heavy-ion storage ring in Heidelberg, Germany. The measured merged beams recombination rate coefficients (MBRRC) for relative energies from 0 to 75 eV are presented, covering all dielectronic recombination (DR) resonances associated with 3s → 3p and 3p → 3d core transitions in the spectroscopic species Fe X and Fe XI, respectively. We compare our experimental results to state-of-the-art multiconfiguration Breit-Pauli (MCBP) calculations and find significant differences. Poor agreement between the measured and theoretical resonance structure is seen for collision energies below 48 eV for Fe X and below 35 eV for Fe XI. The integrated resonance strengths, though, are in reasonable agreement. At higher energies, good agreement is seen for the resonance structure but for the resonance strengths theory is significantly larger than experiment by a factor of 1.5 (2) for Fe X (Fe XI). From the measured MBRRC, we have extracted the DR contributions and transform them into plasma recombination rate coefficients (PRRCs) for astrophysical plasmas with temperatures in the range of 10²-10⁷ K. This range spans across the regimes where each ion forms in photoionized or in collisionally ionized plasmas. For both temperature regimes, the experimental uncertainties are 25% at a 90% confidence level. As expected based on predictions from active galactic nucleus observations as well as our previous laboratory and theoretical work on M-shell iron, the formerly recommended DR data severely underestimated the rate coefficient at temperatures relevant for photoionized gas. At these temperatures relevant for photoionized gas, we find agreement between our experimental results and MCBP theory. This is somewhat surprising given the poor agreement in MBRRC resonance structure. At the higher temperatures relevant for collisionally ionized gas, the MCBP calculations yield an Fe XI DR rate coefficient that is significantly larger than the experimentally derived one. We present parameterized fits to our experimentally derived DR PRRC for ease of inclusion into astrophysical modeling codes.