Purée the basil, oil, and salt, then filter through a fine-mesh sieve into a bowl or measuring cup with a spout, pressing on the solids to get as much flavor as possible into the oil. Discard solids and set oil aside.

Melt butter together with the sesame oil and add minced shallot; cook until the shallot is clear. Add the peas and broth, and cook until the peas are soft — about 5-7 minutes. Do not let them turn olive green. Purée and push the mixture through a fine-mesh sieve into the same pot. Add the soy sauce and lime juice and mix well. Keep warm while you make the quail.

Preheat the oven to 350°F. Combine the stuffing ingredients seasoning well with salt, and stuff into the four quail, tying the legs with kitchen string, then cut the string ends short. Season on both sides with salt and pepper. Place an oven-proof skillet over medium-high heat and, when hot, add the neutral oil. Brown the quail well, starting breast side down, then flip and brown the other side. Transfer the skillet to the oven and roast for 10 minutes (internal temperature of 155°F). Remove quail from the skillet and set on paper towels to drain while you prepare the plates. Cut and remove the strings.

Divide warmed pea purée among four plates (about 1/2 cup per plate) and rotate plates to distribute the sauce. Carefully place a quail on each plate, then drizzle the Thai basil oil over the pea purée; top the quail with a sprig of Thai basil. Serve immediately.

Serves 4.