  We present a theoretical model of composite excitonic states in doped
semiconductors. Many-body interactions between a photoexcited electron-hole
pair and the electron gas are integrated into a computationally tractable
few-body problem, solved by the variational method. We focus on electron-doped
ML-MoSe$_2$ and ML-WSe$_2$ due to the contrasting character of their conduction
bands. In both cases, the core of the composite is a tightly-bound trion (two
electrons and valence-band hole), surrounded by a region depleted of electrons.
The composite in ML-WSe$_2$ further includes a satellite electron with
different quantum numbers. The theory is general and can be applied to
semiconductors with various energy-band properties, allowing one to calculate
their excitonic states and to quantify the interaction with the Fermi sea.
