Preheat oven to 400°F and line a baking sheet with parchment paper.
Clean the portabella caps with a damp cloth and carefully remove the stems and scrape out the dark gills using a spoon.
Place mushroom caps on the baking sheet, gill-side up, and lightly brush with a tiny bit of the olive oil.
In a medium skillet over medium-high heat, add the remaining olive oil, onion, and carrot, sautéing until softened.
Add the ground turkey to the skillet, breaking it up with a spatula and cooking until browned and no longer pink.
Stir in the minced garlic, sea salt, black pepper, and dried oregano, cooking for 1 minute until fragrant.
Pour in the tomato puree, reduce heat to low, and simmer the bolognese for 10 minutes to develop the flavors.
While the sauce simmers, place the mozzarella cheese inside the mushroom caps and roast in the oven for 8-10 minutes until the cheese is bubbly.
Remove mushrooms from the oven, plate them, and generously spoon the turkey bolognese over the melted cheese.
Garnish with freshly chopped basil before serving.