Let $q=p^n>3$ be an odd prime power and let $g:\F_q\to\F_q$ be an even perfect nonlinear (PN) function with $g(0)=0$. We construct almost perfect nonlinear (APN) functions $G_{λ,M}(x)=g(x)+M(x)+λ(x^{q-1}-1)$, where $λ\in\F_q^*$, $-λ\notin g(\F_q)$, and $M$ is an $\F_p$-linearized permutation polynomial. These one-point modifications of $g+M$ have boomerang uniformity one, the smallest possible value for APN functions in odd characteristic. We give complete differential and boomerang spectrum formulas and obtain closed-form evaluations for selected constructions based on Coulter--Matthews monomials and Dembowski--Ostrom polynomials. For $q>7$, comparison of differential spectra establishes CCZ-inequivalence to all APN monomial families listed in the paper and every APN binomial in the original Ness--Helleseth family or its extension by Zeng et al. We characterize CCZ-equivalence within our construction, showing that any two linearly inequivalent even PN functions over the same field yield CCZ-inequivalent APN functions. We obtain three pairwise CCZ-inequivalent APN functions with boomerang uniformity one over $\F_{p^{15\ell}}$ for every $\ell\ge1$, six over $\F_{p^{120r}}$ for every $r\ge1$, and unbounded numbers of such classes in characteristic three. We further prove that CCZ-equivalence and EA-equivalence coincide for functions at Hamming distance at most $m$ from a PN function, provided that $m(m+1)<q-q/p$. We also show that all our APN functions have the maximal possible algebraic degree $n(p-1)$. Finally, we exhibit both CCZ-equivalent and CCZ-inequivalent APN pairs with boomerang uniformity one at Hamming distance one.

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