Desktop: rebuild the hosts board once the typing in the find box pauses, not once a letter

src/DodoSSH.Client.App/Views/HostsScreen.axaml

@@ -230,9 +230,13 @@
 
         Ctrl+K is named on it because the palette is the other way to reach a host by typing, and somebody
         who has found this box should know about the one that also connects on Enter.
+
+        One way, and passed back by the code-behind once the typing pauses: every change to the filter
+        rebuilds the board's sections and their cards, and a name typed at speed used to do that once a
+        letter. The view model still filters the moment it is told. See HostsScreen.OnFilterTyped.
       -->
       <Border Grid.Row="1" Margin="26,0,26,16">
-        <TextBox x:Name="HostFilter" Text="{Binding HostFilter}" Height="40" CornerRadius="10"
+        <TextBox x:Name="HostFilter" Text="{Binding HostFilter, Mode=OneWay}" Height="40" CornerRadius="10"
                  FontFamily="{StaticResource MonoFont}"
                  PlaceholderText="Find a host by name, address or note…   ·   Ctrl+K searches and connects" />
       </Border>

src/DodoSSH.Client.App/Views/HostsScreen.axaml.cs

@@ -106,6 +106,48 @@ internal sealed partial class HostsScreen : UserControl
         Board.AddHandler(ContextRequestedEvent, OnContextRequested, RoutingStrategies.Tunnel);
 
         WireTheBand();
+
+        HostFilter.TextChanged += OnFilterTyped;
+    }
+
+    /// <summary>How long the find box waits for the typing to pause before the board is rebuilt.</summary>
+    private static readonly TimeSpan FilterPause = TimeSpan.FromMilliseconds(150);
+
+    /// <summary>The wait itself, replaceable so a test can say when a pause is over rather than sleep through one.</summary>
+    internal Func<TimeSpan, Task> Pause { get; set; } = static span => Task.Delay(span);
+
+    /// <summary>How many times the find box has changed, so a wait can tell it is no longer the latest.</summary>
+    private int typed;
+
+    /// <summary>Hands the find box's text to the vault once the typing pauses.</summary>
+    /// <remarks>
+    /// <para>
+    /// Every change to the filter rebuilds the board's sections and every card in them, and a name typed at
+    /// speed used to do that once a letter. Now it is done once, with the whole of what was typed.
+    /// </para>
+    /// <para>
+    /// A wait rather than a binding delay, and on the dispatcher's own queue: the box is bound one way, so a
+    /// filter the vault sets itself — cleared, or restored — still lands in the box at once, and the
+    /// change it makes to the box comes back here as the same text, which changes nothing.
+    /// </para>
+    /// </remarks>
+    private void OnFilterTyped(object? sender, TextChangedEventArgs e) =>
+        _ = PassTheFilterOnAsync(HostFilter.Text ?? string.Empty, ++typed);
+
+    private async Task PassTheFilterOnAsync(string text, int keystroke)
+    {
+        await Pause(FilterPause).ConfigureAwait(true);
+
+        // A later letter has its own wait, and it is the one that speaks.
+        if (keystroke != typed)
+        {
+            return;
+        }
+
+        if (Vault is { } vault && !string.Equals(vault.HostFilter, text, StringComparison.Ordinal))
+        {
+            vault.HostFilter = text;
+        }
     }
 
     /// <summary>The band's own wiring, and the two keys that go with a set.</summary>