Natural and engineering materials typically expand when heated, and thermal effects must be considered in the design of any structure subject to significant temperature variations. Materials with zero, negative or more in general tunable thermal expansion would be useful to mitigate undesired effects such as thermal stresses, dimensional changes and warping. Microstructured periodic materials, also called metamaterials, emerged in the last two decades as suitable candidates to achieve fully tunable thermal expansion, both in the positive and negative range, while using virtually any constituent material. The mechanisms of tunable thermal expansion metamaterials rely on the presence, within each cell, of at least two constituent materials with positive, but different, thermal expansion coefficients, and voids. In metaplates, the desired effective thermal expansion coefficient is obtained by a proper design of a two-dimensional geometry of a unit cell, with different parts made of the two constituent materials, which is extruded in the out-of-plane direction. In the present work, we propose a new configuration of the unit cell, where a base structure is made of one constituent material, and the other constituent is layered on top of some parts of the base structure. This configuration, unlike those available in the literature, is suitable for microlithography fabrication processes used, e.g., for micro-electro-mechanical systems. With this new configuration, it is possible to obtain a metaplate with a target thermal expansion coefficient, which can be positive, zero or even negative. The asymmetric layers of material cause in general the deflection of the metamaterial plate, but also this thermal curvature can be tuned by design and, if necessary, it can be set to zero. Parametric studies show the potentialities of the newly proposed metaplate, which can have the desired combination of thermal expansion and curvature.