Energy of propagating electromagnetic waves can be fully absorbed in a thin lossy layer, but only in a narrow frequency band, as follows from the causality principle. On the other hand, it appears that there are no fundamental limitations on broadband matching of thin resonant absorbing layers. However, known thin absorbers produce significant reflections outside of the resonant absorption band. In this paper, we explore possibilities to realize a thin absorbing layer that produces no reflected waves in a very wide frequency range, while the transmission coefficient has a narrow peak of full absorption. Here we show, both theoretically and experimentally, that a thin resonant absorber, invisible in reflection in a very wide frequency range, can be realized if one and the same resonant mode of the absorbing array unit cells is utilized to create both electric and magnetic responses. We test this concept using chiral particles in each unit cell, arranged in a periodic planar racemic array, utilizing chirality coupling in each unit cell but compensating the field coupling at the macroscopic level. We prove that the concept and the proposed realization approach also can be used to create nonreflecting layers for full control of transmitted fields. Our results can have a broad range of potential applications over the entire electromagnetic spectrum including, for example, perfect ultracompact wave filters and selective multifrequency sensors.