Warm vegetable broth in a pot over medium to medium-high heat.
Heat butter and olive oil in a wide pan until butter is melted.
Add the onion and mushrooms and cook over medium to medium-high heat for 5 minutes.
Turn heat to medium and add the rice. Stir until the rice is well coated.
Add one ladleful (approximately ¼ cup) of broth and stir.
When broth is absorbed, add the wine and stir.
Once wine is absorbed, add one ladleful of broth until it is absorbed, stirring constantly.
Continue to add one ladleful of broth at a time to the rice and stir constantly until broth is absorbed and rice is just tender and has a creamy consistency (will take about 20-25 minutes minutes from this point).
Once the rice is creamy and you have used all or most of your broth (you might not need it all), stir in peas, cheese, tarragon, and white pepper. Cook over medium heat for 2 more minutes, stirring constantly.