Clean the portobello caps with a damp paper towel. Gently remove the stems and use a spoon to scrape out the dark gills, creating a nice cavity for the filling.
Whisk together the olive oil, balsamic vinegar, minced garlic, thyme, salt, and pepper in a small bowl. Brush this mixture generously over both sides of the mushroom caps. Let them marinate for at least 10 minutes.
Heat 1 tablespoon of olive oil in a skillet over medium heat. Add the shallot and cook until translucent, about 2 minutes. Add the spinach and cook just until wilted, approximately 1-2 minutes. Remove from heat and let cool slightly.
In a medium bowl, combine the wilted spinach mixture with feta cheese, sun-dried tomatoes, pine nuts, lemon zest, lemon juice, and red pepper flakes if using. Gently mix until combined.
Preheat your grill to medium-high heat (around 375°F). Place the marinated mushroom caps on the grill, gill-side up, and cook for about 5 minutes until they begin to soften. Flip and grill for another 2-3 minutes on the cap side.
Return the mushrooms to the grill gill-side up and carefully fill each cap with the spinach-feta mixture. Close the grill lid and cook for an additional 3-4 minutes, until the cheese starts to melt and the stuffing is heated through.