How to retrieve values of custom / user-defined columns through EF?

Properties marked with [NotMapped] are just ignored by EF Core in all operations, including queries, so even the raw SQL includes such column in the result set, the populated classes by EF Core will just have the default value for the property (in this case, 0 (zero)). The corresponding fluent configuration is even called Ignore.

That's why I'm personally against using [NotMapped] / Ignored members in entity classes - it's so easy to make a mistake to try using them in a LINQ query (for instance, in Where, OrderBy, Select etc.) or like here, and get runtime exceptions or silently skipped data.

So what is the solution. The class you are showing does not really represent a table record, but a view of that table record. Hence it should be a separate class containing the properties returned by the SP.

One way is to create a new class and copy the properties needed. Another (in case you want to reuse all properties of the original class) is to create new class, inherit the original and add the additional properties.

In both cases the new class must be registered in the EF Core model as entity (either keyless or normal) with no associated table.

So let's do the second approach. Remove Lvl property along with the [NotMapped] attribute from MyTreeModel, and add a class similar to this

public class MyTreeModelView : MyTreeModel
{
    public int Lvl { get; set; }
}

and fluent configuration like this

modelBuilder.Entity<MyTreeModelView>(builder =>
{
    builder.HasBaseType((Type)null); // do not consider MyTreeModelView as part of a database inheritance
    builder.ToView(null); // the easiest way to say that it has no associated table
    builder.HasKey(e => e.SubsetId); // or HasNoKey() to treat it as keyless
});

and change the type of the result for SP call

public IQueryable<MyTreeModelView> TreeProcedure(long vSet, DateTime vDate)
{
    return Set<MyTreeModelView>() // <-- this is how you access it w/o explicit `DbSet` property
        .FromSqlRaw("EXEC dbo.tree_structure @vSet, @vDate",
            new SqlParameter("vSet", vSet),
            new SqlParameter("vDate", vDate.ToString("yyyy-MM-dd"))
        );
}

Finally, in case you can replace the SP with TVF, you can utilize EF Core 5.0+ Mapping a queryable function to a table-valued function as shown in the documentation link. But that's only for the TreeProcedure implementation - all other steps (new class, fluent mapping) are still needed.

This is to provide sample for suggestion in @IvanStoev's answer to replace the SP with TVF.

1. Create the function.

CREATE FUNCTION dbo.TreeStructure(@vSet bigint, @vDate datetime) RETURNS TABLE AS
RETURN
(
    WITH Hierarchy
    AS (
        SELECT Id, Date, SetId, SubsetId, PSubsetId, Order, 0 AS Lvl
        FROM [myTable]
        WHERE Set IN (@vSet)
        AND Date = @vDate
        UNION ALL 
            SELECT t2.Id, t2.Date, t2.SetId, t2.SubsetId, t2.PSubsetId, t2.Order, Lvl + 1
            FROM [myTable] t2 
            INNER JOIN hierarchy h 
            ON t2.PSubsetId = h.SubsetId
            AND t2.Date = h.Date 
            AND t2.SetId = h.SetId
    )
    SELECT * FROM Hierarchy
)

2. Add below configuration within OnModelCreating method.

modelBuilder.HasDbFunction(typeof(myDbContext).GetMethod(nameof(TreeProcedure), new[] { typeof(long), typeof(DateTime) }));

3. Change the SP call method to call TVF.

public IQueryable<MyTreeModelView> TreeProcedure(long vSet, DateTime vDate) => FromExpression(() => TreeProcedure(vSet, vDate));

4. To retrieve the dataset.

var result = await _context.TreeProcedure(1, DateTime.Today).ToListAsync();