Preheat the oven to 200 degrees. Remove the stems from the portobellos and gently scrape out the dark gills with a teaspoon; otherwise, they release a lot of moisture. Rub the caps all over with a tablespoon of olive oil, salt, and pepper.
Place the portobellos cup-side down on a baking sheet and roast them for ten minutes first. This allows the moisture released by the mushrooms to drain away instead of pooling in the cap; a stuffed portobello that skips this pre-baking step will swim in its own grey liquid at the table.
Remove the baking sheet from the oven and flip the caps over. Mix the goat cheese with the finely chopped garlic, the leaves from a few sprigs of thyme, and a grind of pepper, then divide the mixture among the cavities. Coarsely break the walnuts and press them into the cheese.
Bake the stuffed portobellos for another ten minutes, until the cheese melts and browns slightly. Drizzle the honey over them in the final minute so it just caramelizes.
Toss the arugula with the remaining olive oil and balsamic vinegar, and place the warm portobellos on top. Finish with some fresh thyme.