Everywhere-$δ$-dense graphs are defined as graphs on $n$ vertices in which every vertex has degree at least $δn$ for some constant $δ> 0$. Approximation schemes are vital for handling NP-hard optimization problems, but for many graph cut problems, existing PTAS algorithms often suffer from running times of $n^{f(1/\varepsilon)}$. In this paper, we bring PTASs down to EPTASs for several fundamental minimization problems on everywhere-$Ω(1)$-dense graphs. Specifically, we present the first Efficient Polynomial-Time Approximation Scheme (EPTAS), running in time $f(1/\varepsilon)n^{O(1)}$, for the ConstrainedMinCut problem under a global constraint on vertex weights, a problem that captures BalancedSeparator and SmallSetExpansion. Moreover, we give the first EPTASs for MinQuotientCut and ProductSparsestCut on everywhere-$δ$-dense graphs with integer-valued dense vertex weights; these problems generalize the four well-known problems UniformSparsestCut, EdgeExpansion, Conductance, and NormalizedCut. Our main technical contribution is an EPTAS for ConstrainedMinCut, based on the weak regularity lemma and sampling and estimation techniques. We then obtain EPTASs for MinQuotientCut and ProductSparsestCut via a unified reduction that invokes this algorithm as a subroutine. In contrast, previous works giving PTASs for these problems on everywhere-$δ$-dense graphs typically rely on powerful tools such as the Lasserre hierarchy or specific integer programming technique, which we avoid.