TL;DR: Blazor Desktop using Avalonia UI currently doesn't work on Linux.
From the current looks of it, there is no easy way to embed a browser view into Avalonia. WebViewControl-Avalonia comes to mind; it does however rely on CefGlue for embedding Chromium but has — as of writing this — no intention of supporting Linux (see OutSystems/CefGlue#18 (comment)).
The repository currently consists of three projects:
src/BlazorDesktop
: A .NET 6 Blazor Server project.src/RazorClassLibrary
: The main Razor components in a reusable class library.src/AvaloniaUI
: The Avalonia MVVM application; it does not do anything useful at the moment.
A possibly useful takeaway here is how to refactor a Blazor Server project into an application and a class library; see the Blazor Desktop - hybrid Web and Desktop apps in .NET 6! YouTube video for the inspiration. It relies on a Windows Forms application to include the web view however, so its proposed solution currently only runs on Windows.
Apart from Avalonia, the original epic on Blazor Desktop support, dotnet/maui#2536, further states that first class Linux support is currently not planned but might be added in the future.