How to provide value for layout's parameters from page?

Consider the layout below:

@inherits LayoutComponentBase
@layout ProductionsLayout

@Body

<div>
    @TrademarkMessage
</div>

@code {
    public string TrademarkMessage { get; set; } = 
        "Doctor Who is a registered trademark of the BBC. " +
        "https://www.doctorwho.tv/";
}

Here the value of TrademarkMessage is provided right in the layout code.

How can I provide it from a page ?

@page "/"
...

You need to use a version of the notification pattern.

First a data class:

public class LayoutData
{
    public string TrademarkMessage { get; private set; } = "My TradeMark Message";

    public event EventHandler? DataChanged;

    public void SetTradeMarkMessage(string message)
    {
        this.TrademarkMessage = message;
        this.DataChanged?.Invoke(this, EventArgs.Empty);
    }
}

the Layout:

@using SO76837241.Data;
@inherits LayoutComponentBase

<PageTitle>SO76837241</PageTitle>

<div class="page">
    <div class="sidebar">
        <NavMenu />
    </div>

    <main>
        <div class="top-row px-4">
            <span>@_layoutData.TrademarkMessage</span>

            <a href="https://docs.microsoft.com/aspnet/" target="_blank">About</a>
        </div>

        <article class="content px-4">
            <CascadingValue Value="_layoutData" IsFixed>
                @Body
            </CascadingValue>
        </article>
    </main>
</div>

@code {
    private LayoutData _layoutData = new();

    protected override void OnInitialized()
    {
        // As this is internal to the component we don't need to de-register
        _layoutData.DataChanged += (object? sender, EventArgs e) => this.StateHasChanged();
    }
}

And Index:

@page "/"
@using SO76837241.Data;

<PageTitle>Index</PageTitle>

<h1>Hello, world!</h1>

Welcome to your new app.

<SurveyPrompt Title="How is Blazor working for you?" />

@code {
    [CascadingParameter] private LayoutData? _layoutData { get; set; }

    protected override void OnInitialized()
    {
        if (_layoutData is not null)
            _layoutData.SetTradeMarkMessage("Total Bull!");
    }
}