  Recent efforts to numerically simulate compact objects in alternative
theories of gravity have largely focused on the time-evolution equations.
Another critical aspect is the construction of constraint-satisfying initial
data with precise control over the properties of the systems under
consideration. Here, we augment the extended conformal thin sandwich framework
to construct quasistationary initial data for black hole systems in scalar
Gauss-Bonnet theory and numerically implement it in the open-source SpECTRE
code. Despite the resulting elliptic system being singular at black hole
horizons, we demonstrate how to construct numerical solutions that extend
smoothly across the horizon. We obtain quasistationary scalar hair
configurations in the test-field limit for black holes with linear/angular
momentum as well as for black hole binaries. For isolated black holes, we
explicitly show that the scalar profile obtained is stationary by evolving the
system in time and compare against previous formulations of scalar Gauss-Bonnet
initial data. In the case of the binary, we find that the scalar hair near the
black holes can be markedly altered by the presence of the other black hole.
The initial data constructed here enables targeted simulations in scalar
Gauss-Bonnet simulations with reduced initial transients.
