For the horseradish cream:
Put the yoghurt in a kitchen towel-lined strainer set over a bowl, set aside to drain and thicken, about 60 minutes.
Discard the watery liquid. Whisk the horseradish into the thickened yoghurt.
For the onion jam:
Heat two tablespoons of the olive oil in a medium frying pan, over medium heat. Add the onions, thyme sprig and season with salt and pepper, to taste. Cook, covered, until the onions have wilted, about 10 to 15 minutes (give them a stir every now and then with a wooden spoon to prevent them from sticking).
Add the red wine, and simmer over high heat until most of the wine is absorbed into the onions. Add the honey and red wine vinegar and simmer gently until the onions are jam-like, about 15 minutes.
For the portobello burgers:
Preheat a griddle pan over a medium-high heat. Whisk the oil, garlic and balsamic vinegar in a bowl. Strip and add the leaves from the sprig of thyme. Coat the mushroom caps all over with the flavoured olive oil.
Griddle the mushrooms, turning as needed, until tender but not mushy, about 3 to 4 minutes per side. Season with salt and pepper.
Sandwich the mushrooms between the baps, top with some of the onion jam, some lettuce, and a dollop of the horseradish cream. Serve warm.