Solar twins in Gaia DR3 GSP-Spec I. Building a large catalog of Solar twins with ages

[Abbreviated] Context. Solar twins, stars whose stellar parameters (Teff, log g, and [M/H]) are very close to the Solar ones, offer a unique opportunity to investigate Galactic archaeology with very high accuracy and precision. However, most previous catalogs of Solar twins contain only a small number of objects (typically a few tens), and their selection functions are poorly characterized. Aims. We aim at building a large catalog of Solar twins from Gaia DR3 GSP-Spec, providing model-driven, rather than data-driven, stellar parameters including ages, together with a well-characterized selection function. Methods. Using stellar parameters from the Gaia DR3 GSP-Spec catalog, we selected Solar-twin candidates whose parameters lie within +- 200 K in Teff, +- 0.2 in log g, and +- 0.1 dex in [M/H] of the Solar values. Candidates unlikely to be genuine Solar twins were removed using Gaia flags and photometric constraints. We determined accurate ages for individual twins with a Bayesian isochrone-projection method, considering three combinations of parameters: Teff, [M/H], and either log g, M_G, or M_Ks. We also constructed a mock catalog to characterize the selection function. Results. Our final GSP-Spec Solar-twin catalog contains 6,594 stars. The mock catalog consisting of 75,588 artificial twins well reproduces the main characteristics of the observed catalog, especially for ages determined with M_G or M_Ks. To demonstrate the usefulness of our catalog, we compared chemical abundances [X/Fe] with age. We statistically confirmed the age--[X/Fe] relations for several species (e.g., Al, Si, Ca, and Y), demonstrating that trends previously identified in small but very high-precision samples persist in a much larger, independent sample. Conclusions. Our study bridges small high-precision Solar-twin samples and large data-driven ones, enabling demographic studies of Solar twins.