First make the caramelized onions: Place the onionsin a large frying pan over medium-high heat, cover andcook, stirring infrequently, until they are dry and almoststicking to the pan (about 20 minutes).
Stir in the oil and a large pinch of salt, then reduce theheat to medium-low and cook, stirring occasionally andadding oil as needed to keep them from sticking withoutgetting greasy, for 40 to 60 minutes, depending on howsilky you want them. Season to taste and set aside.
Meanwhile, brush the steaks all over with a little oliveoil and season lightly with salt and pepper. Place a largegrill pan over medium-high heat and, when hot, add thesteaks and cook for 2 to 3 minutes on each side (or untildone to your liking), then remove from the pan and set aside.
Add the remaining olive oil to the pan, add the mushroomsand cook over high heat for 5 to 10 minutes untilgolden brown.
Add the wine and thyme to the pan, bring to a simmerand cook for 3 to 4 minutes until the wine has reducedslightly.
Plate the steaks with a spoonful of sauce, the caramelizedonions on the side and a few carrots too, if you like.