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private void QueryReportTimerCallback(object state) {
    QueryReportCallbackHandler(this, EventArgs.Empty);
}

private async void QueryReportCallbackHandler(object sender, EventArgs e) {
    if (await semaphoreQueryReportSlim.WaitAsync(10))
        await queryReportsCore();
}

//CTOR
public MyClass() {
    QueryReportCallbackEvent += QueryReportCallbackHandler;
    var QueryReportTimer = new Timer(QueryReportTimerCallback, null, TimeSpan.Zero, TimeSpan.FromSeconds(15));
}

event EventHandler QueryReportCallbackEvent = delegate { };

private void QueryReportTimerCallback(object state) {
    QueryReportCallbackHandler(this, EventArgs.Empty);
}

private async void QueryReportCallbackHandler(object sender, EventArgs e) {
    if (await semaphoreQueryReportSlim.WaitAsync(10))
        await queryReportsCore();
}

private async Task queryReportsCore() {
    try {
        if (machineConfigurations != null) {
            var tasks = machineConfigurations
                            .Where(x => x.QueryReport)
                            .Select(configuration => HandleReport(configuration));

            await Task.WhenAll(tasks);
        }
    } catch (Exception e) {
        logger.LogError(e, e.Message);
    } finally {
        semaphoreQueryReportSlim.Release();
    }
}

private async Task HandleReport(MachineConfiguration configuration) {
    if (configuration.IsConnectionValid) {
        var queryReport = new QueryReport(configuration, ReportConfigurations, fileContainer, applicationConfiguration, logger);
        await QueryAReport(configuration, queryReport);
    }
}

private void QueryReportTimerCallback(object state) {
    QueryReportCallbackEvent(this, EventArgs.Empty);
}

private async void QueryReportCallbackEventHandler(object sender, EventArgs e) {
    if (await semaphoreQueryReportSlim.WaitAsync(10))
        await queryReportsCore();
}

//CTOR
public MyClass() {
    QueryReportCallbackEvent += QueryReportCallbackEventHandler;
    var QueryReportTimer = new Timer(QueryReportTimerCallback, null, TimeSpan.Zero, TimeSpan.FromSeconds(15));
}

event EventHandler QueryReportCallbackEvent = delegate { };

private void QueryReportTimerCallback(object state) {
    QueryReportCallbackEvent(this, EventArgs.Empty);
}

private async void QueryReportCallbackEventHandler(object sender, EventArgs e) {
    if (await semaphoreQueryReportSlim.WaitAsync(10))
        await queryReportsCore();
}

private async Task queryReportsCore() {
    try {
        if (machineConfigurations != null) {
            var tasks = machineConfigurations
                            .Where(x => x.QueryReport)
                            .Select(configuration => HandleReport(configuration));

            await Task.WhenAll(tasks);
        }
    } catch (Exception e) {
        logger.LogError(e, e.Message);
    } finally {
        semaphoreQueryReportSlim.Release();
    }
}

private async Task HandleReport(MachineConfiguration configuration) {
    if (configuration.IsConnectionValid) {
        var queryReport = new QueryReport(configuration, ReportConfigurations, fileContainer, applicationConfiguration, logger);
        await QueryAReport(configuration, queryReport);
    }
}

As already stated in the OP async void should be avoided as much as possible.

The one exception being for event handlers, which can be the loophole to achieving the desired behavior while still having the ability to catch and handle any thrown exceptions.

As already stated in the OP async void should be avoided as much as possible.

The one exception to that rule being for event handlers, which can be the loophole to achieving the desired behavior while still having the ability to catch and handle any thrown exceptions.

Lastly take note of how the functions were broken down into smaller chunks that allowed for cleaner, more easy to read code.