Bring the brown rice and water to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until the rice is tender and the liquid has been absorbed, 45 to 50 minutes.
While rice is cooking, heat 1 tablespoon olive oil in a large skillet over medium heat. Sprinkle chicken breasts with salt and pan-fry in the hot oil until golden brown on both sides and no longer pink in the center, 5 to 8 minutes per side. Set chicken aside.
Heat 2 tablespoons olive oil in the skillet; cook and stir garlic and pepper flakes in hot oil just until fragrant, about 1 minute; stir in spinach and cook, stirring often, until wilted, about 2 more minutes. Sprinkle with salt if desired.
Spread cooked brown rice onto a serving platter and arrange chicken breasts over the rice. Spread spinach over chicken, sprinkle with pine nuts and goat cheese, and squeeze lemon juice over the dish to serve.