Make dough: In a food processor, pulse flour, butter, and salt until butter is pea-size. Add 1/2 cup very cold water and pulse just until dough comes together, adding another tbsp. water if needed. Turn dough onto a work surface and press into a mound. Cut off one-fourth of dough, form both portions into flat disks, wrap airtight, and chill at least 1 hour and up to 2 days.
Place larger piece of dough on a floured surface. Roll into a circle about 14 in. across. Butter bottom and sides of an 8-in. springform pan. Lay dough in pan, pressing on bottom and up sides, and allowing excess dough to hang over edge.
Roll out smaller piece of dough a little bigger than pan, then cut to size of pan. Transfer circle to a plate, discarding scraps. Chill shell and circle until ready to use.
Make filling: Heat oil in a large nonstick pan over medium-high heat. Add garlic and cook, stirring, until fragrant, about 1 minute. Add half the spinach and cook, stirring, until starting to wilt. Add remaining spinach and cook until all the leaves are wilted. Season with 1/2 tsp. salt. Spoon spinach into a strainer and let drain completely, then transfer to a paper towel and blot any excess liquid.
Wipe frying pan with a paper towel. Reduce heat to medium-low. Crack 10 eggs into a bowl and stir with a fork just to break up. Melt butter in pan, then add eggs. Gently scramble eggs, stirring occasionally, until mostly solidified but still very wet (they'll cook more in the torta). Sprinkle with remaining 1/2 tsp. salt and the oregano, then transfer to a plate, spreading eggs evenly. Chill until cool.
Assemble torta: Evenly spread half the eggs in bottom of dough-lined pan. Then evenly lay half the cheese over eggs, cutting cheese if needed to make an even layer. Distribute half the spinach over cheese and sprinkle with half the basil. Cover with half the ham. Use a towel to blot peppers completely dry, then lay them over ham, cutting peppers if needed to make an even layer. The peppers are the middle layer. From this point onward, layer the remaining ingredients in reverse order: first ham, then basil, spinach, cheese, and finally eggs.
Using scissors, trim edge of dough to a 1-in. overhang. Fold dough toward center of torta. Lay reserved dough circle on top, pressing down gently on torta with your hand to even out layers. Press edges of dough with your thumb along inside of pan to seal. Using a knife, cut a small x in center of torta to let steam out when it bakes. Wrap torta in plastic wrap and chill at least 2 hours and as long as overnight.
Preheat oven to 375° with a rack set on lowest level. Whisk remaining egg and brush some over top of torta. Bake until golden brown, about 1 hour. Let cool on a rack at least 1 hour and up to 4 hours.
Run a knife between side of torta and pan, then loosen pan and lift pan off torta.
Note: Nutritional analysis is per serving.