In a small saucepan over very low heat, combine tuna, mushrooms, garlic, and grapeseed oil. Drain any oil from anchovies into pan, then set aside anchovy fillets. Cook, uncovered, until garlic is tender enough to be easily pierced with a skewer, about 30 minutes. Remove from heat, transfer to a bowl, and chill until cold, about 1 hour. Discard mushrooms (or save for another use, like soup).
Meanwhile, put potato in a small saucepan, add enough water to cover, and bring to a simmer over medium heat. Cook until tender enough to be easily pierced with a skewer, 10 to 15 minutes. Drain well and set aside.
Strain chilled tuna mixture, reserving oil. Put tuna mixture in a food processor with cooked potato, anchovies, fish sauce, egg yolks, and lemon juice and purée, scraping down bowl occasionally, until smooth, about 1 minute. With motor running, slowly pour in strained oil and whirl just until very thick (if you overprocess, the sauce will break). Season with salt and pepper.
Make ahead: Up to 3 days, chilled airtight.