Place the beets in a pot and cover with salted water. Bring to a rapid boil, then reduce the heat to a simmer. Cook the beets until they can be pierced with a fork—about 10 to 15 minutes. Drain and set aside until cool enough to handle.
In a bowl, combine the chèvre, milk, pecans, herbs and black pepper. Stir until well combined and set aside.
Slice each beet into 3 or 4 horizontal rounds about ¼- to ½-inch thick. In a small bowl, combine the beet slices with a bit of olive oil, salt, pepper and the lemon juice. Toss to coat evenly.
Using tongs, place the beets slices on a hot grill (avoiding the hottest spots) and cook for about 10 minutes per side. Remove the beets to a plate and let cool.
To assemble the stacks, spread about a 1½ teaspoons of the chèvre mixture on top of one beet slice. Place another beet slice on top, and then repeat until there are two or three layers of cheese and all four stacks are assembled.