Get and Set the default browser on macOS. Also is an example at how to leverage other scripting languages within PowerShell

DefaultBrowser.ps1

# NOTE: This still pops a dialog... I can't seem to figure out how to -Force it to not show.
function Set-DefaultBrowser {
    [CmdletBinding(SupportsShouldProcess = $true)]
    param(
        [Parameter(Mandatory = $true)]
        $BrowserId
    )

    $pythonScript = @"
from LaunchServices import LSSetDefaultHandlerForURLScheme
from LaunchServices import LSSetDefaultRoleHandlerForContentType

# 0x00000002 = kLSRolesViewer
# see https://developer.apple.com/library/mac/#documentation/Carbon/Reference/LaunchServicesReference/Reference/reference.html#//apple_ref/c/tdef/LSRolesMask
LSSetDefaultRoleHandlerForContentType(\"public.html\", 0x00000002, \"$BrowserId\")
LSSetDefaultRoleHandlerForContentType(\"public.xhtml\",0x00000002, \"$BrowserId\")
LSSetDefaultHandlerForURLScheme(\"http\", \"$BrowserId\")
LSSetDefaultHandlerForURLScheme(\"https\", \"$BrowserId\")
"@

    if($PSCmdlet.ShouldProcess("Running Python script", $pythonScript)) {

	python -c $pythonScript
    }
}

function Get-DefaultBrowser {
    [CmdletBinding(SupportsShouldProcess = $true)]
    param()

    $appleScriptPath = "$PSScriptRoot/GetDefaultBrowser.scpt"
    if($PSCmdlet.ShouldProcess("Running AppleScript script", (Get-Content -Raw $appleScriptPath))) {
        osascript $appleScriptPath
    }
}

GetDefaultBrowser.scpt

#!/usr/bin/osascript

# Idk what this is but Stack Overflow said so
tell (system attribute "sysv") to set MacOS_version to it mod 4096 div 16
if MacOS_version is 5 then
    set {a1, a2} to {1, 2}
else
    set {a1, a2} to {2, 1}
end if

set pListpath to (path to preferences as Unicode text) & "com.apple.LaunchServices:com.apple.launchservices.secure.plist"
tell application "System Events"
set launchServicesPList to property list file pListpath
    repeat with i in property list items of property list item 1 of contents of launchServicesPList
        if value of property list item a2 of i is in {"http", "https", "public.html", "public.xhtml"} then
            return the value of the property list item a1 of i
        end if
    end repeat
end tell