Heat your oven to 450º F and place a rack in the lowest position.
Add the baby potatoes to a large pot and cover with cold water. Season well with salt and bring to a boil over high heat. Cook until the potatoes are fork tender, about 12 to 15 minutes, then strain and allow the potatoes to cool slightly. When cool enough to handle, lay the potatoes out onto a sheet pan and, using the bottom of a glass or measuring cup, squash each potato down slightly just so it bursts and flattens a bit. Drizzle with 2 tablespoons of olive oil, season with salt and pepper, and toss gently to evenly coat.
Roast the potatoes on the bottom rack of the oven for 15 minutes. Flip and continue to cook for another 10 minutes or until golden and crisp.
While the potatoes roast, prepare the halibut. Heat a large skillet over medium-high heat and pat the fillets dry with a piece of paper towel. Scatter the flour into a shallow dish, season the fillets well with kosher salt and freshly ground black pepper, and dredge in the flour.
Add the remaining tablespoon of olive oil into the pan along with the butter and allow it to melt together. Tap the excess flour off the halibut fillets and place them in the pan to sear until deeply golden brown on the first side, about three to five minutes. Flip the fillets over and continue to cook for two to three minutes just until the internal temperature reaches 140º F or the fish easily flakes when pierced with a fork.
Remove the halibut to four serving plates to rest while you prepare the gremolata.
Place the parsley, herbs, and garlic onto a cutting board and zest over the lemon. Finely chop everything together until well combined. Season with salt and pepper and transfer to a bowl.
Remove the potatoes from the oven and transfer them to the plates. Cut the lemon in half and juice over the fish and potatoes and top with the fresh herby gremolata, reserving two teaspoons.
Stir reserved gremolata into the mayo and serve on the side. Enjoy!