Lightly toast pecans in a 300 degree oven for 5 to 10 minutes; let cool. Measure out 1/3 cup (40g) of pecans and set aside for later. Line a baking sheet with parchment paper or a silicone baking mat.
Combine sugar, water and cream of tartar in a small, heavy saucepan. Cook over medium heat, stirring occasionally, until sugar is completely dissolved. One sugar starts to boil, cover and let simmer without stirring for 2 minutes. Remove lid. Brush down any stray sugar crystals on the sides of the pan with a pastry brush dipped in water.
Continue to cook without stirring (you can occasionally swirl the pan if needed to ensure it caramelizes evenly) until sugar darkens to a light amber color.
Remove from heat and stir in 1 cup of pecans until coated in melted sugar. Pour onto a prepared baking sheet, breaking up any large chunks (you want individual nuts as separated as possible). Let cool completely.
Once pecans are completely cooled, transfer to a food processor and pulse 2 to 3 times or until coarsely chopped. Ideally you're looking for pea and chickpea-sized pieces here. Transfer to a bowl. Shake the bowl back and forth a bit to let the smaller crumbs sift to the bottom, then pick the larger pieces off of the top. You should end up with about 1/2 cup of larger pieces, and 1/4 cup or so of smaller pieces and crumbs (save these, you'll use them later!)
To make biscotti
Finely grind reserved 1/3 cup of toasted pecans in food processor until very finely ground (you're looking for a pecan meal-like texture; don't grind too much or you'll end up with pecan butter!)
Combine ground pecans, flour, baking powder, salt and cinnamon in a bowl and whisk until evenly combined. Set aside.
Place eggs in a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment. Beat on medium high speed until frothy, then add sugar and continue beating until pale and thick, 2 to 3 minutes on medium-high speed.
Reduce mixer to low and drizzle in melted butter, followed by vanilla and 3 drops of praline flavoring (if using).
Sprinkle flour mixture over top and mix on low speed until almost completely incorporated. Add 1/2 cup of the larger bits of pecan praline and stir until evenly incorporated. Cover and refrigerate for 30 minutes.
Preheat oven to 325 degrees F. Line a baking sheet with parchment paper. If you'd like, you can trace two 2.5-by-10-inch rectangles side by side on the back of the parchment paper as a guide. Leave about 4 inches of space between the two rectangles to allow for spreading.
Divide chilled dough in half and dollop bits of dough in two rows along rectangle guides; lightly flour your hands and pat dough into two 2.5-by-10-inch rectangles, sprinkling with flour as needed so the dough doesn't stick to your hands.
Bake for 25 to 30 minutes, rotating sheet half way through baking, until top is just beginning to crack and bottom is light golden brown. Remove from oven and let cool for 30 minutes.
Transfer biscotti to a cutting board. Using a sharp serrated knife, cut on a bias into 1/2-inch wide pieces. Lay pieces, cut side down, on baking sheet, leaving 1/2-inch of space between each piece.
Return to oven and bake for an additional 15 to 20 minutes, flipping pieces about half-way through, until crisp and golden brown. If you want harder biscotti you can bake bake them as long as 35 minutes (harder biscotti will travel better and keep longer, but they are really too hard eaten plain and are best when dipped in coffee). Transfer to a wire rack and let cool completely.
For white chocolate coating
Finely chop about 3/4 of your white chocolate. Leave the remaining chocolate in 3 or 4 large chunks (set these aside).
Place chopped chocolate and cocoa butter in a double boiler set over a pot of gently simmering water (you can also use a chocolate melting pot if you have one). Melt, stirring occasionally, until chocolate is fully melted and measures 110 degrees F on an instant read thermometer (don't let it get much hotter or it may start to separate). Remove from heat and add reserved chunks of white chocolate. Continue to stir until the chocolate has cooled to 83 degrees F, then briefly return to heat to bring it back to 87 degrees F. At this point you can test the temper by dropping a bit on a piece of parchment paper; it should set up snappily in about 5 minutes. Stir in 3 to 5 drops of praline flavoring oil (do not use extract here or the chocolate may seize).
Dip one end of each cookie into melted white chocolate, shaking off excess. Lay flat on parchment or silicone-lined baking sheet. Sprinkle with a pinch of reserved praline crumbs. Let cool until chocolate is completely set.
Biscotti will keep at room temperature in an airtight container for up to 2 weeks.