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Working in 2 batches, pulse mushrooms in a food processor until finely chopped. Transfer to a bowl. Add walnuts to food processor; pulse until finely chopped. Transfer to a medium bowl. Add onion, carrot, and celery to food processor; process until very finely chopped.
Heat ¼ cup oil in a large skillet over medium-high. Add mushrooms in an even layer; cook, undisturbed, until browned in spots, about 8 minutes. Season with ½ teaspoon salt, toss, and cook for 1 minute. Transfer to bowl with walnuts. Reduce heat to medium. Add remaining 2 tablespoons oil to skillet. Stir in onion mixture and ¼ teaspoon salt; cook, stirring often, until mixture is golden and almost dry, 10 to 12 minutes. Stir in mushrooms and walnuts. Add wine; cook until absorbed, 2 to 3 minutes. Add tomato paste; cook, stirring, until combined, about 5 minutes. Add stock; simmer over low, stirring once or twice, until sauce is very thick, 35 to 40 minutes. Add pepper, remaining ½ teaspoon salt, and half-and-half; cook, stirring, for 2 minutes
Toss pasta with desired amount of sauce. Serve with cheese. Place any remaining sauce in an airtight container; refrigerate for up to 3 days or freeze for up to 3 months.