Solar twins, stars whose stellar parameters (effective temperature Teff, surface gravity logg, and metallicity [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. We aim at building a large catalog of Solar twins from Gaia DR3 GSP- Spec, providing stellar parameters including ages determined using a model-driven, rather than data-driven, method, together with a well-characterized selection function. Using stellar parameters from the Gaia DR3 GSP-Spec catalog, we selected Solar-twin candidates whose parameters lie within +/-200K in Teff, +/-0.2 in logg, and +/-0.1dex 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 logg, M_G_, or M_Ks_. We also constructed a mock catalog to characterize the properties and selection function of our observed sample. Our final GSP-Spec Solar-twin catalog contains 6594 stars. The mock catalog consisting of 75588 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. Our study bridges small high-precision Solar-twin samples and large data-driven ones by providing a large sample with model-driven stellar parameters and a quantified selection function, enabling demographic studies of Solar twins.