Prepare your ingredients. Preheat the oven to 400 F (200 C). Line a baking sheet with parchment paper. Mince the garlic. Slice the sun-dried tomatoes. Thinly slice the leek and rinse well.
Cook the leeks. Melt the butter in a large pan. Add the leeks and ½ teaspoon salt, then saute over medium heat, stirring regularly. After 7 to 8 minutes, they should be tender and browning. At this point, add the garlic. Cook just until fragrant, about 30 seconds, then add the white wine and deglaze the pan by gently scraping off any brown bits.
Prepare the mushrooms. As the leeks cook, hollow out the mushrooms by removing the stems and gills (set aside; don’t discard). Brush the caps with olive oil, and put them on the prepared baking sheet open side up. Bake for about 15 minutes, until tender.
Prepare the filling. As the mushrooms bake, finely mince the mushroom stems and gills. Add to the leeks, along with the sun dried tomatoes, smoked paprika, and cumin. Cook over medium heat until the mushrooms give up their juices and the whole thing starts to get wet, about 5 minutes. Add the rice and stir. If the mix looks very dry, add a little more white wine, up to ¼ cup. Cook the mixture for a couple minutes just to heat through. Taste and add salt if necessary.
Finish the mushrooms. Blot any excess moisture from the mushrooms with a paper towel. Scoop the rice mixture into the mushrooms and top each with 1 ounce of the goat cheese. Return the mushrooms to the oven and bake for about 5 more minutes just to heat the goat cheese through. Sprinkle with smoked paprika, garnish with basil, and enjoy!