Preheat oven to 375°F (190°C).
Cook the quinoa: combine 1/3 cup pre-rinsed dry quinoa and 2/3 cup water in a medium pot. Cover, bring to a boil, then reduce heat to a low simmer and cook until the water is mostly absorbed, about 12–15 minutes. Turn off the heat and let the quinoa sit, covered, for 5 minutes. Uncover and fluff with a fork.
While the quinoa cooks, prepare the mushrooms: remove and discard mushroom stems if present and wipe the caps clean with a damp cloth or paper towel. Place the 4 portobello caps gill-side up on a rimmed baking sheet.
Brush the mushroom caps with 1 tablespoon olive oil and evenly drizzle with 2 tablespoons balsamic vinegar. Season the caps with kosher or sea salt to taste.
Bake the prepared mushrooms in the preheated oven for 10 minutes. After 10 minutes, remove the baking sheet and carefully drain any excess liquid from the mushroom caps.
Make the filling: in a medium mixing bowl combine the cooked quinoa, 1/2 teaspoon black pepper, 1/4 teaspoon crushed red pepper flakes, kosher or sea salt to taste, 1 cup diced vine‑ripe tomatoes, 1/2 cup whole‑grain bread crumbs, 1/4 cup freshly chopped basil, and 1/2 cup fat‑free feta. Mix until evenly combined.
Divide the quinoa mixture evenly among the 4 mushroom caps (about 1/2 cup filling per mushroom).
Return the stuffed mushrooms to the oven and bake for 12 minutes, or until the mushrooms are tender and the feta is warmed/melted to your liking.
Remove from the oven and serve immediately, or allow to cool slightly before serving.