Preheat the oven to 400ºF and set a wire rack on a baking sheet. Place the mushrooms open side up on the wire rack. Drizzle the mushrooms with olive oil and season with salt and pepper. Use your hands to massage the oil and seasonings into the mushrooms.
Place the mushrooms into the oven and bake them for 12-15 minutes or until they begin to soften and release moisture.
Remove the mushrooms from the oven and let them rest while you prepare the filling.
Heat a large skillet over medium/high heat and add olive oil to the skillet. When the olive oil is fragrant, add the onion to the skillet with a pinch of salt and sauté the onion for 2-3 minutes. Add the garlic and saute for an additional minute.
Add the minced walnuts to the skillet along with the salt, pepper, paprika, and thyme. Toss until all the ingredients are coated in spices. Cook the walnuts over medium heat until they begin to brown (4-5 minutes).
Add the cherry tomatoes to the skillet and let them sauté with the other ingredients for 2-3 minutes.
Remove the skillet from heat and carefully add about ¼ cup of the walnut filling to the middle of each well of portobello mushrooms. The mixture should be a mound of filling. If your mushrooms don’t have a well in the middle, that’s okay! You can still scoop the filling on top of the mushroom.
Turn the heat on the oven to broil.
Crumble goat cheese over the top of each mushroom and place the mushrooms back into the oven for 2-3 minutes or until the cheese has melted. Make sure to check the mushrooms after 1 minute as each oven is different.
Remove the mushrooms from the oven and drizzle each mushroom with pesto. Serve and enjoy.