Cook the broccoli, cauliflower, and carrots separately until tender: Put each vegetable in a medium microwave-proof dish; add about ¼ cup water. Cover tightly and microwave on high for about 5 minutes, or until tender. (Or cook the vegetables in a stovetop steamer for about 5 to 6 minutes.) Drain well and let cool.

While the vegetables are cooking, heat the oil in a small nonstick skillet over medium heat. Add the onion and garlic; cook, stirring occasionally, until the onion is tender, about 4 minutes. Set aside.

For the broccoli layer, put the cooked broccoli into a food processor. Add 1/3 of the onion mixture, 1/3 cup rice, 1 egg, the Parmesan cheese, basil, 1 tablespoon lemon juice, ¼ teaspoon pepper, and 1/8 teaspoon salt. Process until smooth. Spread evenly in the bottom of the prepared pan; smooth the surface.

For the cauliflower layer, put the cooked cauliflower into a food processor. Add 1/3 of the onion mixture, 1/3 cup rice, 1 egg, 1 tablespoon lemon juice,1/8 teaspoon salt, and red pepper flakes. Process until smooth. Spread evenly over the broccoli layer; smooth the surface.

For the carrot layer, put the carrots into a food processor. Add the remaining 1/3 of the onion mixture, 1/3 cup rice, 1 egg, 1 tablespoon lemon juice, ¼ teaspoon pepper, and 1/8 teaspoon salt, and dill. Process until smooth. Spread evenly over the cauliflower layer; smooth the surface. Cover with a piece of lightly oiled wax paper.

Place the loaf pan in a roasting pan and pour in boiling water to come about 1 inch up the sides of the pans. Bake for about 1 hour 15 minutes, or until the top of the patés feels firm to the touch and a knife inserted in the center comes out clean. Remove from the oven, remove the wax paper on top, and let cool in the pan on a wire rack.

To make the sauce, meanwhile, stir together the ingredients in a small bowl. Taste and adjust the seasoning.

To serve, loosen the edges of the patés with a knife. Turn out of the pan onto a serving platter and peel off any waxed paper that adheres.

Cut the room temperature patés into ¾-inch-thick slices. Transfer to small plates, top each serving with about 1 tablespoon of the sauce, and garnish. Provide forks for your guests.

Advance preparation

The sauce will keep in the refrigerator for up to 2 days in a covered container. The paté can be made early the day it is to be served; cover with plastic wrap and refrigerate. Bring the sauce and patés to room temperature for serving.