Place beef roast in a large pan (at least 8-quart size); cover with the water.
Heat to simmering.
Simmer for 10 minutes; add pork and bone, bay leaves, and whole peppers.
Bring to simmer again; skim foam from surface.
Cover; simmer slowly until meats are just tender, 2 1/2 to 3 hours, adding salt after meats have cooked for 1 hour.
Prepare and reserve vegetables: Wash potatoes, and peel a 1-inch band around center of each.
Wash cabbage; trim and discard outer leaves, leaving as many of the firm, inner green leaves as possible.
Trim end of core; cut cabbage into 6 or 8 wedges.
Peel and quarter the turnips.
About 30 minutes before meat is done, add the turnips and potatoes; cook until just tender. (You can tie turnips and potatoes separately in cheesecloth for easy removal later.) Cook peas as directed on package until just tender; then drain and reserve.
Remove turnips, potatoes, beef, and pork from simmering liquid.
Arrange the vegetables around the meat on a large, deep platter; cover platter with foil and keep warm.
Strain the cooking liquid through several layers of cheesecloth.
Remove 2 cups of the strained broth; reserve.
Return remaining broth to pan.
Add cabbage; simmer for 5 to 7 minutes or until just tender.
Remove from cooking liquid; arrange on meat platter.
Meanwhile, put half of the cooked peas into blender container with 1 cup of the reserved broth; whirl until smooth.
Press pureed peas through fine wire strainer, discarding pulp.
Repeat with remaining peas and reserved broth.
Stir strained peas into broth in pan; simmer until heated.