  The study of fractional Chern insulators and their exotic anyonic excitations
poses a major challenge in current experimental and theoretical research.
Quantum simulators, in particular ultracold atoms in optical lattices, provide
a promising platform to realize, manipulate, and understand such systems with a
high degree of controllability. Recently, an atomic $\nu=1/2$ Laughlin state
has been realized experimentally for a small system of two particles on 4 by 4
sites. The next challenge concerns the preparation of Laughlin states in
extended systems, ultimately giving access to anyonic braiding statistics or
gapless chiral edge-states in systems with open boundaries. Here, we propose
and analyze an experimentally feasible scheme to grow larger Laughlin states by
connecting multiple copies of the already existing 4-by-4-system. First, we
present a minimal setting obtained by coupling two of such patches, producing
an extended 8-by-4-system with four particles. Then, we analyze different
preparation schemes, setting the focus on two shapes for the extended system,
and discuss their respective advantages: While growing strip-like lattices
could give experimental access to the central charge, square-like geometries
are advantageous for creating quasi-hole excitations in view of braiding
protocols. We highlight the robust quantization of the fractional quasi-hole
charge upon using our preparation protocol. We benchmark the performance of our
patchwork preparation scheme by comparing it to a protocol based on coupling
one-dimensional chains. We find that the patchwork approach consistently gives
higher target-state fidelities, especially for elongated systems. The results
presented here pave the way towards near-term implementations of extended
Laughlin states in quantum gas microscopes and the subsequent exploration of
exotic properties of topologically ordered systems in experiments.
