To the best of my knowledge, the Prophet ﷺ did not take his meals in a big tray at all, nor did he ever eat well-baked thin bread, nor did he ever eat at a dining table.
مَا عَلِمْتُ النَّبِيَّ ﷺ أَكَلَ عَلَى سُكُرُّجَةٍ قَطُّ وَلاَ خُبِزَ لَهُ مُرَقَّقٌ قَطُّ وَلاَ أَكَلَ عَلَى خِوَانٍ قِيلَ لِقَتَادَةَ فَعَلَى مَا كَانُوا يَأْكُلُونَ قَالَ عَلَى السُّفَرِ