Place a small plate in the freezer for later use. Chop the carrots into 1-inch pieces. Place in a food processor and pulse until the carrots are approximately the size of rice, about 15-20 pulses.
Combine the carrots, sugar, and the zest and juice of one lemon and orange zest in a medium sized saucepan. Place the pan over medium heat and bring to a simmer. Reduce heat to medium-low and cook, stirring occasionally, for 30 minutes.
Remove the plate from the freezer and place two teaspoons of jam onto the plate. Freeze for exactly one minute. Remove and drag your finger through the jam. If the jam begins to wrinkle, remove the jam from the heat. If not, place the plate back in the freezer and continue cooking the jam. Repeat the freezer test every 5 minutes until it wrinkles. Remove jam from the heat, transfer to a heatproof bowl, and refrigerate for at least 1 hour.
Preheat the oven to 375°F. Slice the baguette on the bias (diagonally) into ¼ inch thick slices. Place on a parchment covered baking sheet. Brush with butter and sprinkle with salt and pepper. Transfer to the oven and bake until lightly toasted, about 8-10 minutes.
Meanwhile, tear the prosciutto into 1½-2 inch pieces. Place on another parchment lined baking sheet and bake for 10-15 minutes, or until crisp. Slice the burrata into 1 inch pieces.
To assemble, place a slice of burrata on top of each baguette slice and sprinkle with salt and pepper. Top with a dollop (about 1 teaspoon) of carrot jam. Place one of the crisp prosciutto pieces into the jam. Arrange a few sprouts of the microgreens on top and serve.