Heat the grill pan. While waiting for the grill to be ready, make the sauce. Heat 1 tablespoon of the olive oil in a medium saucepan over medium-high heat until hot. Add the shallot and cook briefly until soft and light brown. Add 1 teaspoon of the thyme and stir, then add the red wine. Bring to a boil and cook until reduced to about 1/4-cup. Add the broth, return to a boil and continue to cook until reduced to about 3/4-cup. Season with salt and pepper.
Measure 2 tablespoons sauce into a medium bowl and whisk in the red wine vinegar. While whisking rapidly, drizzle in 1 tablespoon of the olive oil. Season with salt and pepper. Set the vinaigrette aside until ready to assemble the dish. Whisk the butter into the remaining sauce, season with salt and pepper to taste and keep warm.
In a medium bowl, whisk together the remaining 1/2-cup olive oil, the balsamic vinegar, the garlic and the remaining thyme. Holding the mushrooms one at a time over the bowl, spoon the marinade over them and let the excess run back into the bowl. Repeat until the mushrooms have been well coated all over Lay the mushrooms on a plate and season both sides with salt and pepper
Brush or wipe a little olive oil on the grill. Place the mushrooms, gill side down, on the grill and cook, turning once, until browned on both sides and cooked through (about 4 minutes total). Remove and keep warm. Add the spinach to the vinaigrette and toss well. Divide among 4 plates. Top with the mushrooms, gill side up, and spoon the sauce onto the mushrooms.