In a medium saucepan, heat a drizzle of oil over medium-high. Add the rice and cook, stirring often, until toasted and golden brown, about 5 minutes. Add the stock; bring to a boil. Reduce the heat to low, then cover and cook until the rice is tender, 15 to 18 minutes. Remove from the heat and keep covered until you serve.
In a food processor, pulse all the pesto ingredients until a thick sauce forms.
Heat a round-bottom skillet or pot over high. Add 1 tbsp. grapeseed oil, one turn of the pan. Add the mushrooms and cook, stirring constantly, until browned and crispy at the edges, about 8 minutes; transfer to a plate. Add 1 tbsp. oil, one turn of the pan. Add half the vegetables. Stir-fry until the vegetables are crisp-tender, 3 to 5 minutes; transfer to a plate. Repeat with the remaining oil and vegetables. Return all the vegetables to the skillet. Add the tamari, sesame oil and mushrooms. Remove from the heat, add the pesto and toss to coat.
Serve the stir-fry on a bed of rice, topped with the toasted sesame seeds.