It provides to monolog handlers for AppInsights: for tracking dependency (AppInsightsDependencyHandler) and traces (AppInsightsTraceHandler).
It might be tricky to find a proper moment to flush everything to AppInsights for long running processes (e.g: consumers). One of the solution might be a Buffer Handler. You can easily wrap both handlers by Buffer Handler and set up an overflow buffer. They are designed to flush everything into AppInsight's when a buffer overflows.
Size of the telemetry is limited to 64 kilobytes. Handlers checks the length of the telemetry before adding it to the AppInsights Client's queue. If it exceeds the limit it won't be added! It means that AppInsights should not be the only source for your data. You should always have a copy somewhere else where you don't have such limitations.