We introduce a family of averaged meta-Fibonacci recursions $$ Q_{α,m}(n) = 1+ \left\lfloor α\frac1m \sum_{j=1}^m Q_{α,m}(n-Q_{α,m}(n-j)) \right\rfloor , $$ with initial conditions $$ Q_{α,m}(1)=\cdots=Q_{α,m}(m)=1. $$ Unlike classical Hofstadter-type recursions, the averaging mechanism produces highly regular large-scale behavior. For the critical parameter value $α=1$, we prove global well-definedness for all $m\ge1$, establish an exact triangular block structure, and show that the value $k$ occurs exactly $k$ consecutive times. As a consequence, $$ Q_{1,m}(n)\sim \sqrt{2n}. $$ For the supercritical regime $α>1$, we derive an asymptotic slope constraint showing that any positive linear growth rate, if it exists, must equal $$ 1-α^{-1}. $$ Numerical experiments support the existence of a linear-growth phase and suggest a broader universality phenomenon for generalized averaging operators, including positive-power $L^p$-means. These results indicate that averaging induces a robust regularization mechanism for self-referential recursive systems, leading to stable slow-growth dynamics and nontrivial phase structure.