We suggest a criterion to minimize the amount of fine-tuning in a composite Higgs model. The paradigm of this type of model is the top-condensate model of Bardeen, Hill and Lindner (BHL). Although "minimally fine-tuned", this model failed to account correctly for the masses of the top quark and the 125~GeV Higgs boson. We propose a generalization of the BHL model that employs finely-tuned extended technicolor (ETC) plus technicolor (TC) interactions. The additional freedom of this model may accommodate both m_t(173) and M_H(125). This paper studies the large-N_{TC} and N_C limit of this model in which technicolor is weak and does not contribute to electroweak symmetry breaking. Refinements including walking-TC dynamics and a renormalization group analysis of m_t and M_H will appear in a subsequent paper. A likely generic signal of this model is enhanced production of dibosons (W^+W^- and W^\pm Z^0) and tribosons (W^+W^- Z^0) via rho_T^{0,\pm} and omega_T decay.

21 pages. Version to be published in Physical Review D (modulo a few PRD style diktats); paragraph on S-parameter and its status in walking technicolor models added to Sec. 4 at referee's request; typos fixed in Appendix equations; reference added